Theorem sbceq1g 3027

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 3023	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2		csbconstg 3021	. . 3 |- ( A e. V -> [_ A / x ]_ C = C )
3	2	eqeq2d 2152	. 2 |- ( A e. V -> ( [_ A / x ]_ B = [_ A / x ]_ C <-> [_ A / x ]_ B = C ) )
4	1, 3	bitrd 187	1 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   [.wsbc 2913   [_csb 3007

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  f1od2  6140