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Theorem issubgoilem 20976
Description: Lemma for issubgoi 20977. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
issubgoilem.1  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )
Assertion
Ref Expression
issubgoilem  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
Distinct variable groups:    x, A, y    y, B    x, G, y    x, H, y    x, Y, y
Allowed substitution hint:    B( x)

Proof of Theorem issubgoilem
StepHypRef Expression
1 oveq1 5865 . . 3  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
2 oveq1 5865 . . 3  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
31, 2eqeq12d 2297 . 2  |-  ( x  =  A  ->  (
( x H y )  =  ( x G y )  <->  ( A H y )  =  ( A G y ) ) )
4 oveq2 5866 . . 3  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
5 oveq2 5866 . . 3  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
64, 5eqeq12d 2297 . 2  |-  ( y  =  B  ->  (
( A H y )  =  ( A G y )  <->  ( A H B )  =  ( A G B ) ) )
7 issubgoilem.1 . 2  |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )
83, 6, 7vtocl2ga 2851 1  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858
This theorem is referenced by:  issubgoi  20977
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-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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