Theorem sbciegf 2986

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
sbciegf.1	|- F/ x ps
sbciegf.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbciegf	|- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. 2 |- F/ x ps
2		sbciegf.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen 1442	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		sbciegft 2985	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
5	1, 3, 4	mp3an23 1324	1 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   F/wnf 1453   e. wcel 2141   [.wsbc 2955

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by:  sbcieg  2987  iunxsngf  3950  opelopabf  4259  eqerlem  6544