Theorem r19.42v 2766

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 2765	. 2 |- (E.x e. A (ps /\ ph) <-> (E.x e. A ps /\ ph))
2		ancom 437	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
3	2	rexbii 2640	. 2 |- (E.x e. A (ph /\ ps) <-> E.x e. A (ps /\ ph))
4		ancom 437	. 2 |- ((ph /\ E.x e. A ps) <-> (E.x e. A ps /\ ph))
5	1, 3, 4	3bitr4i 268	1 |- (E.x e. A (ph /\ ps) <-> (ph /\ E.x e. A ps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by:  ceqsrexbv  2974  ceqsrex2v  2975  2reuswap  3039  2reu5  3045  iunrab  4014  iunin2  4031  iundif2  4034  addcass  4416  elxp2  4803  cnvuni  4896  f1oiso  5500