Theorem r19.36av 1763

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 1762	. 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 1746	. . 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 960  A.wral 1648  E.wrex 1649

This theorem is referenced by:  iinss 2604  uniimadom 4820  fsequb2 6525  lmuni 7948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-ral 1652  df-rex 1653

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