Theorem r19.42v 2688

Description: Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.42v	|- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem r19.42v

Step	Hyp	Ref	Expression
1		r19.41v 2687	. 2 |- ( E. x e. A ( ps /\ ph ) <-> ( E. x e. A ps /\ ph ) )
2		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	rexbii 2537	. 2 |- ( E. x e. A ( ph /\ ps ) <-> E. x e. A ( ps /\ ph ) )
4		ancom 266	. 2 |- ( ( ph /\ E. x e. A ps ) <-> ( E. x e. A ps /\ ph ) )
5	1, 3, 4	3bitr4i 212	1 |- ( E. x e. A ( ph /\ ps ) <-> ( ph /\ E. x e. A ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wrex 2509

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-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-rex 2514

This theorem is referenced by:  ceqsrexbv  2934  ceqsrex2v  2935  2reuswapdc  3007  iunrab  4013  iunin2  4029  iundif2ss  4031  iunopab  4370  elxp2  4737  cnvuni  4908  elunirn  5890  f1oiso  5950  oprabrexex2  6275  genpdflem  7694  1idprl  7777  1idpru  7778  ltexprlemm  7787  rexuz2  9776  4fvwrd4  10336  divalgb  12436