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Theorem ixpeq1d 7962
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 7961 . 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 1523  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-fn 5929  df-ixp 7951
This theorem is referenced by:  elixpsn  7989  ixpsnf1o  7990  dfac9  8996  prdsval  16162  isfunc  16571  funcpropd  16607  natfval  16653  natpropd  16683  dprdval  18448  ptval  21421  dfac14  21469  ptuncnv  21658  ptunhmeo  21659  hoidmvle  41135  hoimbl  41166
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