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Theorem ixpeq2dv 6832
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 451 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32ixpeq2dva 6831 1  |-  ( ph  -> 
X_ x  e.  A  B  =  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   X_cixp 6817
This theorem is referenced by:  prdsval  13355  brssc  13691  isfunc  13738  natfval  13820  isnat  13821
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-ral 2548  df-in 3159  df-ss 3166  df-ixp 6818
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