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Theorem 2rmorex 3140
Description: Double restricted quantification with "at most one," analogous to 2moex 2354. (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 2574 . . 3  |-  F/_ y A
2 nfre1 2764 . . 3  |-  F/ y E. y  e.  B  ph
31, 2nfrmo 2885 . 2  |-  F/ y E* x  e.  A E. y  e.  B  ph
4 rspe 2769 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
54ex 425 . . . . 5  |-  ( y  e.  B  ->  ( ph  ->  E. y  e.  B  ph ) )
65ralrimivw 2792 . . . 4  |-  ( y  e.  B  ->  A. x  e.  A  ( ph  ->  E. y  e.  B  ph ) )
7 rmoim 3135 . . . 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 16 . . 3  |-  ( y  e.  B  ->  ( E* x  e.  A E. y  e.  B  ph 
->  E* x  e.  A ph ) )
98com12 30 . 2  |-  ( E* x  e.  A E. y  e.  B  ph  ->  ( y  e.  B  ->  E* x  e.  A ph ) )
103, 9ralrimi 2789 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 1726   A.wral 2707   E.wrex 2708   E*wrmo 2710
This theorem is referenced by:  2reu2  27943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rmo 2715
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