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Theorem proplem 13592
Description: Lemma for mndpropd 14398. (Contributed by Mario Carneiro, 6-Dec-2014.)
Hypothesis
Ref Expression
proplem.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x F y )  =  ( x G y ) )
Assertion
Ref Expression
proplem  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  B ) )  -> 
( X F Y )  =  ( X G Y ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    ph, x, y   
y, Y    x, G, y    x, X, y
Allowed substitution hint:    Y( x)

Proof of Theorem proplem
StepHypRef Expression
1 proplem.1 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x F y )  =  ( x G y ) )
21ralrimivva 2635 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 oveq1 5865 . . . 4  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
4 oveq1 5865 . . . 4  |-  ( x  =  X  ->  (
x G y )  =  ( X G y ) )
53, 4eqeq12d 2297 . . 3  |-  ( x  =  X  ->  (
( x F y )  =  ( x G y )  <->  ( X F y )  =  ( X G y ) ) )
6 oveq2 5866 . . . 4  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
7 oveq2 5866 . . . 4  |-  ( y  =  Y  ->  ( X G y )  =  ( X G Y ) )
86, 7eqeq12d 2297 . . 3  |-  ( y  =  Y  ->  (
( X F y )  =  ( X G y )  <->  ( X F Y )  =  ( X G Y ) ) )
95, 8rspc2v 2890 . 2  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y )  ->  ( X F Y )  =  ( X G Y ) ) )
102, 9mpan9 455 1  |-  ( (
ph  /\  ( X  e.  A  /\  Y  e.  B ) )  -> 
( X F Y )  =  ( X G Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem is referenced by:  mndpropd  14398  grpidpropd  14399  grpsubpropd2  14567  cmnpropd  15098  rngpropd  15372  lmodprop2d  15687  lsspropd  15774  lmhmpropd  15826  lbspropd  15852  assapropd  16067  asclpropd  16085  psrplusgpropd  16313  phlpropd  16559  gsumpropd2lem  23379
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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