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Theorem r19.12 2825
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 2574 . . . 4  |-  F/_ y A
2 nfra1 2761 . . . 4  |-  F/ y A. y  e.  B  ph
31, 2nfrex 2766 . . 3  |-  F/ y E. x  e.  A  A. y  e.  B  ph
4 ax-1 6 . . 3  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  ( y  e.  B  ->  E. x  e.  A  A. y  e.  B  ph ) )
53, 4ralrimi 2792 . 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 2771 . . . . 5  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
76com12 31 . . . 4  |-  ( y  e.  B  ->  ( A. y  e.  B  ph 
->  ph ) )
87reximdv 2822 . . 3  |-  ( y  e.  B  ->  ( E. x  e.  A  A. y  e.  B  ph 
->  E. x  e.  A  ph ) )
98ralimia 2784 . 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 setvar class
Syntax hints:    -> wi 4    e. wcel 1756   A.wral 2710   E.wrex 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716
This theorem is referenced by:  iuniin  4176  ucncn  19835  ftc1a  21484  rngoid  23821  rngmgmbs4  23855  heicant  28379
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