Theorem ixpeq2dva 8941

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
ixpeq2dva.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
ixpeq2dva	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dva

Step	Hyp	Ref	Expression
1		ixpeq2dva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
2	1	ralrimiva 3136	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8940	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
4	2, 3	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  Xcixp 8926

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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-ss 3964  df-ixp 8927

This theorem is referenced by:  ixpeq2dv  8942  dfac9  10179  xpsrnbas  17586  funcpropd  17922  natpropd  18001  prdsmgp  20134  frlmip  21776  elptr2  23569  dfac14  23613  xkoptsub  23649  prdsxmslem2  24529  rrxip  25409  ptrest  37320  prdsbnd2  37496  hoidmvlelem3  46218  ovnhoilem1  46222  ovnhoilem2  46223  hoicoto2  46226  ovnlecvr2  46231  ovncvr2  46232  ovnovollem1  46277  ovnovollem2  46278  hoimbl2  46286  vonhoire  46293  iccvonmbllem  46299  vonioolem2  46302  vonicclem2  46305  vonn0ioo2  46311  vonn0icc2  46313