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Theorem proplem 13870
Description: Lemma for mndpropd 14676. (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 2758 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 oveq1 6047 . . . 4  |-  ( x  =  X  ->  (
x F y )  =  ( X F y ) )
4 oveq1 6047 . . . 4  |-  ( x  =  X  ->  (
x G y )  =  ( X G y ) )
53, 4eqeq12d 2418 . . 3  |-  ( x  =  X  ->  (
( x F y )  =  ( x G y )  <->  ( X F y )  =  ( X G y ) ) )
6 oveq2 6048 . . . 4  |-  ( y  =  Y  ->  ( X F y )  =  ( X F Y ) )
7 oveq2 6048 . . . 4  |-  ( y  =  Y  ->  ( X G y )  =  ( X G Y ) )
86, 7eqeq12d 2418 . . 3  |-  ( y  =  Y  ->  (
( X F y )  =  ( X G y )  <->  ( X F Y )  =  ( X G Y ) ) )
95, 8rspc2v 3018 . 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 456 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 359    = wceq 1649    e. wcel 1721   A.wral 2666  (class class class)co 6040
This theorem is referenced by:  mndpropd  14676  grpidpropd  14677  grpsubpropd2  14845  cmnpropd  15376  rngpropd  15650  lmodprop2d  15961  lsspropd  16048  lmhmpropd  16100  lbspropd  16126  assapropd  16341  asclpropd  16359  psrplusgpropd  16584  phlpropd  16841  gsumpropd2lem  24173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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