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Theorem ixpeq2dv 7014
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dv.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
ixpeq2dv  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem ixpeq2dv
StepHypRef Expression
1 ixpeq2dv.1 . . 3  |-  ( ph  ->  B  =  C )
21adantr 452 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32ixpeq2dva 7013 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   X_cixp 6999
This theorem is referenced by:  prdsval  13605  brssc  13941  isfunc  13988  natfval  14070  isnat  14071  dprdval  15488  elpt  17525  elptr  17526  dfac14  17571
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-in 3270  df-ss 3277  df-ixp 7000
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