Theorem r19.42v 2663

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 2662	. 2 |- ( E. x e. A ( ps /\ ph ) <-> ( E. x e. A ps /\ ph ) )
2		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3	2	rexbii 2513	. 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 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-rex 2490

This theorem is referenced by:  ceqsrexbv  2904  ceqsrex2v  2905  2reuswapdc  2977  iunrab  3975  iunin2  3991  iundif2ss  3993  iunopab  4329  elxp2  4694  cnvuni  4865  elunirn  5837  f1oiso  5897  oprabrexex2  6217  genpdflem  7622  1idprl  7705  1idpru  7706  ltexprlemm  7715  rexuz2  9704  4fvwrd4  10264  divalgb  12269