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Theorem 2rmorex 2969
Description: Double restricted quantification with "at most one," analogous to 2moex 2214. (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
StepHypRef Expression
1 nfcv 2419 . . 3  |-  F/_ y A
2 nfre1 2599 . . 3  |-  F/ y E. y  e.  B  ph
31, 2nfrmo 2715 . 2  |-  F/ y E* x  e.  A E. y  e.  B  ph
4 rspe 2604 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
54ex 423 . . . . 5  |-  ( y  e.  B  ->  ( ph  ->  E. y  e.  B  ph ) )
65ralrimivw 2627 . . . 4  |-  ( y  e.  B  ->  A. x  e.  A  ( ph  ->  E. y  e.  B  ph ) )
7 rmoim 2964 . . . 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 ) )
86, 7syl 15 . . 3  |-  ( y  e.  B  ->  ( E* x  e.  A E. y  e.  B  ph 
->  E* x  e.  A ph ) )
98com12 27 . 2  |-  ( E* x  e.  A E. y  e.  B  ph  ->  ( y  e.  B  ->  E* x  e.  A ph ) )
103, 9ralrimi 2624 1  |-  ( E* x  e.  A E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   E.wrex 2544   E*wrmo 2546
This theorem is referenced by:  2reu2  27965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rmo 2551
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