Theorem sbceq2g 3019

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 3013	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2		csbconstg 3011	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eqeq1d 2146	. 2 |- ( A e. V -> ( [_ A / x ]_ B = [_ A / x ]_ C <-> B = [_ A / x ]_ C ) )
4	1, 3	bitrd 187	1 |- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   [.wsbc 2904   [_csb 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  csbsng  3579  f1od2  6125