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Theorem r19.12 2779
Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.12  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  A. y  e.  B  E. x  e.  A  ph )
Distinct variable groups:    x, y    y, A    x, B
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem r19.12
StepHypRef Expression
1 nfcv 2540 . . . 4  |-  F/_ y A
2 nfra1 2716 . . . 4  |-  F/ y A. y  e.  B  ph
31, 2nfrex 2721 . . 3  |-  F/ y E. x  e.  A  A. y  e.  B  ph
4 ax-1 5 . . 3  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  ( y  e.  B  ->  E. x  e.  A  A. y  e.  B  ph ) )
53, 4ralrimi 2747 . 2  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  A. y  e.  B  E. x  e.  A  A. y  e.  B  ph )
6 rsp 2726 . . . . 5  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
76com12 29 . . . 4  |-  ( y  e.  B  ->  ( A. y  e.  B  ph 
->  ph ) )
87reximdv 2777 . . 3  |-  ( y  e.  B  ->  ( E. x  e.  A  A. y  e.  B  ph 
->  E. x  e.  A  ph ) )
98ralimia 2739 . 2  |-  ( A. y  e.  B  E. x  e.  A  A. y  e.  B  ph  ->  A. y  e.  B  E. x  e.  A  ph )
105, 9syl 16 1  |-  ( E. x  e.  A  A. 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 1721   A.wral 2666   E.wrex 2667
This theorem is referenced by:  iuniin  4063  ucncn  18268  ftc1a  19874  rngoid  21924  rngmgmbs4  21958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672
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