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Theorem ixpeq1d 8949
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ixpeq1d (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq1d
StepHypRef Expression
1 ixpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ixpeq1 8948 . 2 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
31, 2syl 17 1 (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Xcixp 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-fn 6564  df-ixp 8938
This theorem is referenced by:  elixpsn  8977  ixpsnf1o  8978  dfac9  10177  prdsval  17500  isfunc  17909  funcpropd  17947  natfval  17994  natpropd  18024  dprdval  20023  ptval  23578  dfac14  23626  ptuncnv  23815  ptunhmeo  23816  hoidmvle  46615  hoimbl  46646
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