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Theorem ixpeq2dva 8951
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dva.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dva (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dva
StepHypRef Expression
1 ixpeq2dva.1 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
21ralrimiva 3144 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
3 ixpeq2 8950 . 2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
42, 3syl 17 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  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-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-ss 3980  df-ixp 8937
This theorem is referenced by:  ixpeq2dv  8952  dfac9  10175  xpsrnbas  17618  funcpropd  17954  natpropd  18033  prdsmgp  20169  frlmip  21816  elptr2  23598  dfac14  23642  xkoptsub  23678  prdsxmslem2  24558  rrxip  25438  ptrest  37606  prdsbnd2  37782  hoidmvlelem3  46553  ovnhoilem1  46557  ovnhoilem2  46558  hoicoto2  46561  ovnlecvr2  46566  ovncvr2  46567  ovnovollem1  46612  ovnovollem2  46613  hoimbl2  46621  vonhoire  46628  iccvonmbllem  46634  vonioolem2  46637  vonicclem2  46640  vonn0ioo2  46646  vonn0icc2  46648
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