Theorem r19.37av 1758

Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.37av	|- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))

Distinct variable group:   ph,x

Proof of Theorem r19.37av

Step	Hyp	Ref	Expression
1		r19.35 1756	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
2		ax-1 4	. . . 4 |- (ph -> (x e. A -> ph))
3	2	r19.21aiv 1710	. . 3 |- (ph -> A.x e. A ph)
4	3	imim1i 16	. 2 |- ((A.x e. A ph -> E.x e. A ps) -> (ph -> E.x e. A ps))
5	1, 4	sylbi 199	1 |- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))

Syntax hints:   -> wi 3   e. wcel 956  A.wral 1642  E.wrex 1643

This theorem is referenced by:  rcla4edv 1875  rcla4devOLD 10367  dtt2 10498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-ral 1646  df-rex 1647

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