Theorem r19.12 2728

Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof 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

Allowed substitution hints:   ph(x,y)   A(x)   B(y)

Proof of Theorem r19.12

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 |- F/_yA
2		nfra1 2665	. . . 4 |- F/yA.y e. B ph
3	1, 2	nfrex 2670	. . 3 |- F/yE.x e. A A.y e. B ph
4		ax-1 6	. . 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	ralrimi 2696	. 2 |- (E.x e. A A.y e. B ph -> A.y e. B E.x e. A A.y e. B ph)
6		rsp 2675	. . . . 5 |- (A.y e. B ph -> (y e. B -> ph))
7	6	com12 27	. . . 4 |- (y e. B -> (A.y e. B ph -> ph))
8	7	reximdv 2726	. . 3 |- (y e. B -> (E.x e. A A.y e. B ph -> E.x e. A ph))
9	8	ralimia 2688	. 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 15	1 |- (E.x e. A A.y e. B ph -> A.y e. B E.x e. A ph)

Syntax hints:   -> wi 4   e. wcel 1710  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621

This theorem is referenced by:  iuniin  3980