Theorem csbeq2gVD 31462

Description: Virtual deduction proof of csbeq2gOLD 31091. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2gOLD 31091 is csbeq2gVD 31462 without virtual deductions and was automatically derived from csbeq2gVD 31462.

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 31120	. . . 4 |- (. A e. V ->. A e. V ).
2		spsbc 3196	. . . 4 |- ( A e. V -> ( A. x B = C -> [. A / x ]. B = C ) )
3	1, 2	e1_ 31183	. . 3 |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ).
4		sbceqg 3674	. . . 4 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
5	1, 4	e1_ 31183	. . 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 31244	. 2 |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ).
9	8	in1 31117	1 |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1362   = wceq 1364   e. wcel 1761   [.wsbc 3183   [_csb 3285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-sbc 3184  df-csb 3286  df-vd1 31116

This theorem is referenced by: (None)