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Theorem prodeq1d 25313
Description: Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)
Hypothesis
Ref Expression
prodeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
prodeq1d  |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 prodeq1 25306 . 2  |-  ( A  =  B  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )
31, 2syl 15 1  |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   prod_cprd 25298
This theorem is referenced by:  prodeq123d  25316
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-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-un 3157  df-if 3566  df-prod 25299
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