Theorem sbceq2g 3160

Description: Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Assertion

Ref	Expression
sbceq2g	|- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)   V( x)

Proof of Theorem sbceq2g

Step	Hyp	Ref	Expression
1		sbceqg 3154	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2		csbconstg 3152	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eqeq1d 2241	. 2 |- ( A e. V -> ( [_ A / x ]_ B = [_ A / x ]_ C <-> B = [_ A / x ]_ C ) )
4	1, 3	bitrd 188	1 |- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   [.wsbc 3042   [_csb 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sbc 3043  df-csb 3139

This theorem is referenced by:  csbsng  3750  f1od2  6431