MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2rmorex Unicode version

Theorem 2rmorex 2982
Description: Double restricted quantification with "at most one," analogous to 2moex 2227. (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 2432 . . 3  |-  F/_ y A
2 nfre1 2612 . . 3  |-  F/ y E. y  e.  B  ph
31, 2nfrmo 2728 . 2  |-  F/ y E* x  e.  A E. y  e.  B  ph
4 rspe 2617 . . . . . 6  |-  ( ( y  e.  B  /\  ph )  ->  E. y  e.  B  ph )
54ex 423 . . . . 5  |-  ( y  e.  B  ->  ( ph  ->  E. y  e.  B  ph ) )
65ralrimivw 2640 . . . 4  |-  ( y  e.  B  ->  A. x  e.  A  ( ph  ->  E. y  e.  B  ph ) )
7 rmoim 2977 . . . 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 2637 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 1696   A.wral 2556   E.wrex 2557   E*wrmo 2559
This theorem is referenced by:  2reu2  28068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rmo 2564
  Copyright terms: Public domain W3C validator