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Theorem r19.12 2439
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 2194 . . . 4  |-  F/_ y A
2 nfra1 2372 . . . 4  |-  F/ y A. y  e.  B  ph
31, 2nfrexxy 2378 . . 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 2407 . 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 2386 . . . . 5  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
76com12 30 . . . 4  |-  ( y  e.  B  ->  ( A. y  e.  B  ph 
->  ph ) )
87reximdv 2437 . . 3  |-  ( y  e.  B  ->  ( E. x  e.  A  A. y  e.  B  ph 
->  E. x  e.  A  ph ) )
98ralimia 2399 . 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 14 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 1409   A.wral 2323   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329
This theorem is referenced by:  iuniin  3695
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