Theorem csbeq2gVD 33425

Description: Virtual deduction proof of csbeq2gOLD 33055. 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 33055 is csbeq2gVD 33425 without virtual deductions and was automatically derived from csbeq2gVD 33425.

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 33084	. . . 4 |- (. A e. V ->. A e. V ).
2		spsbc 3326	. . . 4 |- ( A e. V -> ( A. x B = C -> [. A / x ]. B = C ) )
3	1, 2	e1a 33146	. . 3 |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ).
4		sbceqg 3811	. . . 4 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
5	1, 4	e1a 33146	. . 3 |- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ).
6		imbi2 324	. . . 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 224	. . 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 33207	. 2 |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ).
9	8	in1 33081	1 |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1381   = wceq 1383   e. wcel 1804   [.wsbc 3313   [_csb 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-sbc 3314  df-csb 3421  df-vd1 33080

This theorem is referenced by: (None)