Theorem r19.12 1737

Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers.

Assertion

Ref	Expression
r19.12	|- (E.x e. A A.y e. B ph -> A.y e. B E.x e. A ph)

Distinct variable groups:   x,y   y,A   x,B

Proof of Theorem r19.12

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 |- (x e. A -> A.y x e. A)
2		hbra1 1684	. . . 4 |- (A.y e. B ph -> A.yA.y e. B ph)
3	1, 2	hbrex 1685	. . 3 |- (E.x e. A A.y e. B ph -> A.yE.x e. A A.y e. B ph)
4		ax-1 4	. . 3 |- (E.x e. A A.y e. B ph -> (y e. B -> E.x e. A A.y e. B ph))
5	3, 4	r19.21ai 1709	. 2 |- (E.x e. A A.y e. B ph -> A.y e. B E.x e. A A.y e. B ph)
6		ra4 1691	. . . . . 6 |- (A.y e. B ph -> (y e. B -> ph))
7	6	com12 11	. . . . 5 |- (y e. B -> (A.y e. B ph -> ph))
8	7	a1d 12	. . . 4 |- (y e. B -> (x e. A -> (A.y e. B ph -> ph)))
9	8	r19.22dv 1734	. . 3 |- (y e. B -> (E.x e. A A.y e. B ph -> E.x e. A ph))
10	9	r19.20i 1701	. 2 |- (A.y e. B E.x e. A A.y e. B ph -> A.y e. B E.x e. A ph)
11	5, 10	syl 10	1 |- (E.x e. A A.y e. B ph -> A.y e. B E.x e. A ph)

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

This theorem is referenced by:  iuniin 2568  ringid 8097

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976

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

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