Theorem sbceq2g 3791

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 3785	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2		csbconstg 3388	. . 3 |- ( A e. V -> [_ A / x ]_ B = B )
3	2	eqeq1d 2464	. 2 |- ( A e. V -> ( [_ A / x ]_ B = [_ A / x ]_ C <-> B = [_ A / x ]_ C ) )
4	1, 3	bitrd 261	1 |- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1455   e. wcel 1898   [.wsbc 3279   [_csb 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376

This theorem is referenced by:  csbsng  4042  csbmpt12  4749  f1od2  28358  bj-snsetex  31602  csbmpt22g  31777  csbfinxpg  31825  poimirlem26  32011  cdlemkid3N  34545  cdlemkid4  34546  brtrclfv2  36364  frege116  36620