Theorem csbeq2gVD 39128

Description: Virtual deduction proof of csbeq2gOLD 38765. 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. csbeq2gOLD 38765 is csbeq2gVD 39128 without virtual deductions and was automatically derived from csbeq2gVD 39128.

1::	|- (. A e. V ->. A e. V ).
2:1:	|- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ).
3:1:	|- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ).
4:2,3:	|- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ).
qed:4:	|- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbeq2gVD	|- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Proof of Theorem csbeq2gVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . 4 |- (. A e. V ->. A e. V ).
2		spsbc 3448	. . . 4 |- ( A e. V -> ( A. x B = C -> [. A / x ]. B = C ) )
3	1, 2	e1a 38852	. . 3 |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ).
4		sbceqg 3984	. . . 4 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
5	1, 4	e1a 38852	. . 3 |- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ).
6		imbi2 338	. . . 4 |- ( ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) -> ( ( A. x B = C -> [. A / x ]. B = C ) <-> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) )
7	6	biimpcd 239	. . 3 |- ( ( A. x B = C -> [. A / x ]. B = C ) -> ( ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) )
8	3, 5, 7	e11 38913	. 2 |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ).
9	8	in1 38787	1 |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533

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-11 2034  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-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-vd1 38786

This theorem is referenced by: (None)