Theorem r19.12 2204

Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (The proof was shortened by Andrew Salmon, 30-May-2011.)

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 1317	. . . 4 |- (x e. A -> A.y x e. A)
2		hbra1 2147	. . . 4 |- (A.y e. B ph -> A.yA.y e. B ph)
3	1, 2	hbrex 2149	. . 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 2174	. 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 2155	. . . . 5 |- (A.y e. B ph -> (y e. B -> ph))
7	6	com12 14	. . . 4 |- (y e. B -> (A.y e. B ph -> ph))
8	7	reximdv 2202	. . 3 |- (y e. B -> (E.x e. A A.y e. B ph -> E.x e. A ph))
9	8	ralimia 2166	. 2 |- (A.y e. B E.x e. A A.y e. B ph -> A.y e. B E.x e. A ph)
10	5, 9	syl 12	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 1300  A.wral 2105  E.wrex 2106

This theorem is referenced by:  iuniin 3264  iuniinOLD 3265  ringid 9469

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324

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

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