Theorem sbcim2g 38748

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3477. sbcim2g 38748 is sbcim2gVD 39111 without virtual deductions and was automatically derived from sbcim2gVD 39111 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcim2g	|- ( A e. V -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )

Proof of Theorem sbcim2g

Step	Hyp	Ref	Expression
1		sbcimg 3477	. . . 4 |- ( A e. V -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) <-> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
2	1	biimpd 219	. . 3 |- ( A e. V -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
3		sbcimg 3477	. . 3 |- ( A e. V -> ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
4		imbi2 338	. . . 4 |- ( ( [. 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 ) ) ) )
5	4	biimpcd 239	. . 3 |- ( ( [. 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 ) ) ) )
6	2, 3, 5	syl6ci 71	. 2 |- ( A e. V -> ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )
7		idd 24	. . . 4 |- ( A e. V -> ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) )
8		biimpr 210	. . . 4 |- ( ( [. A / x ]. ( ps -> ch ) <-> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( ( [. A / x ]. ps -> [. A / x ]. ch ) -> [. A / x ]. ( ps -> ch ) ) )
9	3, 7, 8	ee13 38710	. . 3 |- ( A e. V -> ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) ) )
10	9, 1	sylibrd 249	. 2 |- ( A e. V -> ( ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) ) )
11	6, 10	impbid 202	1 |- ( A e. V -> ( [. 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

This theorem is referenced by:  trsbc  38750  trsbcVD  39113