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Theorem ixpssixp 42102
 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 417 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3145 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 ss2ixp 8493 . 2 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1786   ∈ wcel 2112  ∀wral 3071   ⊆ wss 3859  Xcixp 8480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-v 3412  df-in 3866  df-ss 3876  df-ixp 8481 This theorem is referenced by:  ioosshoi  43675  iinhoiicclem  43679  iinhoiicc  43680  iunhoiioo  43682  vonioolem2  43687  vonicclem2  43690
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