Theorem sbceq1g 3121

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

Assertion

Ref	Expression
sbceq1g	|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = C ) )

Distinct variable group:   x, C

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

Proof of Theorem sbceq1g

Step	Hyp	Ref	Expression
1		sbceqg 3117	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2		csbconstg 3115	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eqeq2d 2219	. 2 |- ( A e. V -> ( [_ A / x ]_ B = [_ A / x ]_ C <-> [_ A / x ]_ B = C ) )
4	1, 3	bitrd 188	1 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   [.wsbc 3005   [_csb 3101

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by:  f1od2  6344