Theorem r19.29r 1757

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.

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 1756	. 2 |- ((A.x e. A ps /\ E.x e. A ph) -> E.x e. A (ps /\ ph))
2		ancom 435	. 2 |- ((E.x e. A ph /\ A.x e. A ps) <-> (A.x e. A ps /\ E.x e. A ph))
3		ancom 435	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
4	3	rexbii 1668	. 2 |- (E.x e. A (ph /\ ps) <-> E.x e. A (ps /\ ph))
5	1, 2, 4	3imtr4 219	1 |- ((E.x e. A ph /\ A.x e. A ps) -> E.x e. A (ph /\ ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wral 1645  E.wrex 1646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-ral 1649  df-rex 1650

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