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Theorem ixpssixp 39790
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 397 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3106 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 ss2ixp 8075 . 2 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wnf 1856  wcel 2145  wral 3061  wss 3723  Xcixp 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-in 3730  df-ss 3737  df-ixp 8063
This theorem is referenced by:  ioosshoi  41403  iinhoiicclem  41407  iinhoiicc  41408  iunhoiioo  41410  vonioolem2  41415  vonicclem2  41418
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