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Theorem ixpeq1d 7386
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 7385 . 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 1370   X_cixp 7374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-fn 5530  df-ixp 7375
This theorem is referenced by:  elixpsn  7413  ixpsnf1o  7414  dfac9  8417  prdsval  14513  isfunc  14894  funcpropd  14930  natfval  14976  natpropd  15006  dprdval  16608  dprdvalOLD  16610  ptval  19276  dfac14  19324  ptuncnv  19513  ptunhmeo  19514
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