Theorem r19.29r 2756

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
r19.29r	|- ((E.x e. A ph /\ A.x e. A ps) -> E.x e. A (ph /\ ps))

Proof of Theorem r19.29r

Step	Hyp	Ref	Expression
1		r19.29 2755	. 2 |- ((A.x e. A ps /\ E.x e. A ph) -> E.x e. A (ps /\ ph))
2		ancom 437	. 2 |- ((E.x e. A ph /\ A.x e. A ps) <-> (A.x e. A ps /\ E.x e. A ph))
3		ancom 437	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
4	3	rexbii 2640	. 2 |- (E.x e. A (ph /\ ps) <-> E.x e. A (ps /\ ph))
5	1, 2, 4	3imtr4i 257	1 |- ((E.x e. A ph /\ A.x e. A ps) -> E.x e. A (ph /\ ps))

Syntax hints:   -> wi 4   /\ wa 358  A.wral 2615  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-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  2reu5  3045