Theorem 2rmorex 3412

Description: Double restricted quantification with "at most one," analogous to 2moex 2543. (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 2764	. . 3 |- F/_ y A
2		nfre1 3005	. . 3 |- F/ y E. y e. B ph
3	1, 2	nfrmo 3115	. 2 |- F/ y E* x e. A E. y e. B ph
4		rmoim 3407	. . 3 |- ( A. x e. A ( ph -> E. y e. B ph ) -> ( E* x e. A E. y e. B ph -> E* x e. A ph ) )
5		rspe 3003	. . . . 5 |- ( ( y e. B /\ ph ) -> E. y e. B ph )
6	5	ex 450	. . . 4 |- ( y e. B -> ( ph -> E. y e. B ph ) )
7	6	ralrimivw 2967	. . 3 |- ( y e. B -> A. x e. A ( ph -> E. y e. B ph ) )
8	4, 7	syl11 33	. 2 |- ( E* x e. A E. y e. B ph -> ( y e. B -> E* x e. A ph ) )
9	3, 8	ralrimi 2957	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 1990   A.wral 2912   E.wrex 2913   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rmo 2920

This theorem is referenced by:  2reu2  41187