Theorem r19.42v 2702

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 2701	. 2 |- ( E. x e. A ( ps /\ ph ) <-> ( E. x e. A ps /\ ph ) )
2		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	rexbii 2551	. 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 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-rex 2528

This theorem is referenced by:  ceqsrexbv  2950  ceqsrex2v  2951  2reuswapdc  3023  iunrab  4041  iunin2  4057  iundif2ss  4059  iunopab  4402  elxp2  4769  cnvuni  4943  elunirn  5941  f1oiso  6001  oprabrexex2  6325  genpdflem  7824  1idprl  7907  1idpru  7908  ltexprlemm  7917  rexuz2  9916  4fvwrd4  10478  divalgb  12615