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Theorem ixpssixp 45070
Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
ixpssixp.1 𝑥𝜑
ixpssixp.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
ixpssixp (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)

Proof of Theorem ixpssixp
StepHypRef Expression
1 ixpssixp.1 . . 3 𝑥𝜑
2 ixpssixp.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝐶)
32ex 412 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3256 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 ss2ixp 8946 . 2 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wral 3060  wss 3950  Xcixp 8933
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-12 2177  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-ss 3967  df-ixp 8934
This theorem is referenced by:  ioosshoi  46657  iinhoiicclem  46661  iinhoiicc  46662  iunhoiioo  46664  vonioolem2  46669  vonicclem2  46672
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