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Theorem r19.12 2627
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 2392 . . . 4  |-  F/_ y A
2 nfra1 2564 . . . 4  |-  F/ y A. y  e.  B  ph
31, 2nfrex 2569 . . 3  |-  F/ y E. x  e.  A  A. y  e.  B  ph
4 ax-1 7 . . 3  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  ( y  e.  B  ->  E. x  e.  A  A. y  e.  B  ph ) )
53, 4ralrimi 2595 . 2  |-  ( E. x  e.  A  A. y  e.  B  ph  ->  A. y  e.  B  E. x  e.  A  A. y  e.  B  ph )
6 ra4 2574 . . . . 5  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
76com12 29 . . . 4  |-  ( y  e.  B  ->  ( A. y  e.  B  ph 
->  ph ) )
87reximdv 2625 . . 3  |-  ( y  e.  B  ->  ( E. x  e.  A  A. y  e.  B  ph 
->  E. x  e.  A  ph ) )
98ralimia 2587 . 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 17 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 6    e. wcel 1621   A.wral 2516   E.wrex 2517
This theorem is referenced by:  iuniin  3856  ftc1a  19311  rngoid  20975  rngmgmbs4  21009
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-rex 2521
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