MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.12 Unicode version

Theorem r19.12 2656
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 2419 . . . 4  |-  F/_ y A
2 nfra1 2593 . . . 4  |-  F/ y A. y  e.  B  ph
31, 2nfrex 2598 . . 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 2624 . 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 2603 . . . . 5  |-  ( A. y  e.  B  ph  ->  ( y  e.  B  ->  ph ) )
76com12 27 . . . 4  |-  ( y  e.  B  ->  ( A. y  e.  B  ph 
->  ph ) )
87reximdv 2654 . . 3  |-  ( y  e.  B  ->  ( E. x  e.  A  A. y  e.  B  ph 
->  E. x  e.  A  ph ) )
98ralimia 2616 . 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 15 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 1684   A.wral 2543   E.wrex 2544
This theorem is referenced by:  iuniin  3915  ftc1a  19384  rngoid  21050  rngmgmbs4  21084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
  Copyright terms: Public domain W3C validator