Theorem r19.36av 1760

Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty.

Assertion

Ref	Expression
r19.36av	|- (E.x e. A (ph -> ps) -> (A.x e. A ph -> ps))

Distinct variable group:   ps,x

Proof of Theorem r19.36av

Step	Hyp	Ref	Expression
1		r19.35 1759	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
2		idd 61	. . . 4 |- (x e. A -> (ps -> ps))
3	2	r19.23aiv 1743	. . 3 |- (E.x e. A ps -> ps)
4	3	imim2i 17	. 2 |- ((A.x e. A ph -> E.x e. A ps) -> (A.x e. A ph -> ps))
5	1, 4	sylbi 199	1 |- (E.x e. A (ph -> ps) -> (A.x e. A ph -> ps))

Syntax hints:   -> wi 3   e. wcel 958  A.wral 1645  E.wrex 1646

This theorem is referenced by:  iinss 2600  uniimadom 4810  fsequb2 6524  lmuni 7951

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-ral 1649  df-rex 1650

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