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Theorem ixpeq1d 7400
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ixpeq1d  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem ixpeq1d
StepHypRef Expression
1 ixpeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ixpeq1 7399 . 2  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
31, 2syl 16 1  |-  ( ph  -> 
X_ x  e.  A  C  =  X_ x  e.  B  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399   X_cixp 7388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-fn 5499  df-ixp 7389
This theorem is referenced by:  elixpsn  7427  ixpsnf1o  7428  dfac9  8429  prdsval  14862  isfunc  15270  funcpropd  15306  natfval  15352  natpropd  15382  dprdval  17147  dprdvalOLD  17149  ptval  20156  dfac14  20204  ptuncnv  20393  ptunhmeo  20394
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