Theorem r19.42v 2461

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 2460	. 2 |- (E.x e. A (ps /\ ph) <-> (E.x e. A ps /\ ph))
2		ancom 253	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
3	2	rexbii 2325	. 2 |- (E.x e. A (ph /\ ps) <-> E.x e. A (ps /\ ph))
4		ancom 253	. 2 |- ((ph /\ E.x e. A ps) <-> (E.x e. A ps /\ ph))
5	1, 3, 4	3bitr4i 201	1 |- (E.x e. A (ph /\ ps) <-> (ph /\ E.x e. A ps))

Syntax hints:   /\ wa 97   <-> wb 98  E.wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-rex 2306

This theorem is referenced by:  ceqsrexbv  2669  ceqsrex2v  2670  2reuswapdc  2737  iunrab  3695  iunin2  3711  iundif2ss  3713  iunopab  4009  elxp2  4306  cnvuni  4464  elunirn  5348  f1oiso  5408  oprabrexex2  5699  genpdflem  6490  1idprl  6566  1idpru  6567  ltexprlemm  6574  rexuz2  8300  4fvwrd4  8767