Theorem 2rmorex 3041

Description: Double restricted quantification with "at most one," analogous to 2moex 2275. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2rmorex	|- (E*x e. A E.y e. B ph -> A.y e. B E*x e. A ph)

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

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

Proof of Theorem 2rmorex

Step	Hyp	Ref	Expression
1		nfcv 2490	. . 3 |- F/_yA
2		nfre1 2671	. . 3 |- F/yE.y e. B ph
3	1, 2	nfrmo 2787	. 2 |- F/yE*x e. A E.y e. B ph
4		rspe 2676	. . . . . 6 |- ((y e. B /\ ph) -> E.y e. B ph)
5	4	ex 423	. . . . 5 |- (y e. B -> (ph -> E.y e. B ph))
6	5	ralrimivw 2699	. . . 4 |- (y e. B -> A.x e. A (ph -> E.y e. B ph))
7		rmoim 3036	. . . 4 |- (A.x e. A (ph -> E.y e. B ph) -> (E*x e. A E.y e. B ph -> E*x e. A ph))
8	6, 7	syl 15	. . 3 |- (y e. B -> (E*x e. A E.y e. B ph -> E*x e. A ph))
9	8	com12 27	. 2 |- (E*x e. A E.y e. B ph -> (y e. B -> E*x e. A ph))
10	3, 9	ralrimi 2696	1 |- (E*x e. A E.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  E*wrmo 2618

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-rmo 2623

This theorem is referenced by: (None)