Theorem r19.12 2416

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 2175	. . . 4 |- F/_yA
2		nfra1 2349	. . . 4 |- F/yA.y e. B ph
3	1, 2	nfrexxy 2355	. . 3 |- F/yE.x e. A A.y e. B ph
4		ax-1 5	. . 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 2384	. 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 2363	. . . . 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 2414	. . 3 |- (y e. B -> (E.x e. A A.y e. B ph -> E.x e. A ph))
9	8	ralimia 2376	. 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 14	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 1390  A.wral 2300  E.wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306

This theorem is referenced by:  iuniin  3658