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Theorem ixpeq1d 8948
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 8947 . 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 1537  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-fn 6566  df-ixp 8937
This theorem is referenced by:  elixpsn  8976  ixpsnf1o  8977  dfac9  10175  prdsval  17502  isfunc  17915  funcpropd  17954  natfval  18001  natpropd  18033  dprdval  20038  ptval  23594  dfac14  23642  ptuncnv  23831  ptunhmeo  23832  hoidmvle  46556  hoimbl  46587
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