Theorem r19.37av 2233

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 2231	. 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 2175	. . 3 |- (ph -> A.x e. A ph)
4	3	imim1i 19	. 2 |- ((A.x e. A ph -> E.x e. A ps) -> (ph -> E.x e. A ps))
5	1, 4	sylbi 216	1 |- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))

Syntax hints:   -> wi 3   e. wcel 1300  A.wral 2105  E.wrex 2106

This theorem is referenced by:  rcla4edv 2383  ssiun 3293  dtt2 14951  prtlem10 16265

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-ral 2109  df-rex 2110

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