Theorem sbcim2gVD 39111

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3477. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 38748 is sbcim2gVD 39111 without virtual deductions and was automatically derived from sbcim2gVD 39111.

1::	|- (. A e. B ->. A e. B ).
2::	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
3:1,2:	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
4:1:	|- (. A e. B ->. ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
5:3,4:	|- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
6:5:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
7::	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
8:4,7:	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
9:1:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) ).
10:8,9:	|- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
11:10:	|- (. A e. B ->. ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) ).
12:6,11:	|- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
qed:12:	|- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcim2gVD	|- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

Proof of Theorem sbcim2gVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . 6 |- (. A e. B ->. A e. B ).
2		idn2 38838	. . . . . 6 |- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
3		sbcimg 3477	. . . . . . 7 |- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
4	3	biimpd 219	. . . . . 6 |- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
5	1, 2, 4	e12 38951	. . . . 5 |- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
6		sbcimg 3477	. . . . . 6 |- ( A e. B -> ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7	1, 6	e1a 38852	. . . . 5 |- (. A e. B ->. ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
8		imbi2 338	. . . . . 6 |- ( ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )
9	8	biimpcd 239	. . . . 5 |- ( ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) -> ( ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )
10	5, 7, 9	e21 38957	. . . 4 |- (. A e. B ,. [. A / x ]. ( ph -> ( ps -> ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
11	10	in2 38830	. . 3 |- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
12		idn2 38838	. . . . . 6 |- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ).
13		biimpr 210	. . . . . . 7 |- ( ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( ( [. A / x ]. ps -> [. A / x ]. ch ) -> [. A / x ]. ( ps -> ch ) ) )
14	13	imim2d 57	. . . . . 6 |- ( ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
15	7, 12, 14	e12 38951	. . . . 5 |- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ).
16	1, 3	e1a 38852	. . . . 5 |- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) ).
17		biimpr 210	. . . . . 6 |- ( ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) -> ( ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) )
18	17	com12 32	. . . . 5 |- ( ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) -> ( ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) )
19	15, 16, 18	e21 38957	. . . 4 |- (. A e. B ,. ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ->. [. A / x ]. ( ph -> ( ps -> ch ) ) ).
20	19	in2 38830	. . 3 |- (. A e. B ->. ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) ).
21		impbi 198	. . 3 |- ( ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) -> ( ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ) )
22	11, 20, 21	e11 38913	. 2 |- (. A e. B ->. ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) ).
23	22	in1 38787	1 |- ( A e. B -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)