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Theorem ixpeq2d 45008
Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ixpeq2d.1 𝑥𝜑
ixpeq2d.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2d (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)

Proof of Theorem ixpeq2d
StepHypRef Expression
1 ixpeq2d.1 . . 3 𝑥𝜑
2 ixpeq2d.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ex 412 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3255 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 ixpeq2 8950 . 2 (∀𝑥𝐴 𝐵 = 𝐶X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wnf 1780  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-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  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:  hoicvrrex  46512  ovnlecvr  46514  ovnhoilem1  46557  hoi2toco  46563  ovnlecvr2  46566  opnvonmbllem1  46588
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