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Theorem ixpssixp 44980
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 3253 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 ss2ixp 8943 . 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 1778  wcel 2104  wral 3057  wss 3963  Xcixp 8930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-12 2173  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-ss 3980  df-ixp 8931
This theorem is referenced by:  ioosshoi  46575  iinhoiicclem  46579  iinhoiicc  46580  iunhoiioo  46582  vonioolem2  46587  vonicclem2  46590
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