Theorem ixpeq2dva 8908

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 3144	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8907	. 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 1539   ∈ wcel 2104  ∀wral 3059  Xcixp 8893

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-in 3954  df-ss 3964  df-ixp 8894

This theorem is referenced by:  ixpeq2dv  8909  dfac9  10133  xpsrnbas  17521  funcpropd  17855  natpropd  17933  prdsmgp  20045  frlmip  21552  elptr2  23298  dfac14  23342  xkoptsub  23378  prdsxmslem2  24258  rrxip  25138  ptrest  36790  prdsbnd2  36966  hoidmvlelem3  45611  ovnhoilem1  45615  ovnhoilem2  45616  hoicoto2  45619  ovnlecvr2  45624  ovncvr2  45625  ovnovollem1  45670  ovnovollem2  45671  hoimbl2  45679  vonhoire  45686  iccvonmbllem  45692  vonioolem2  45695  vonicclem2  45698  vonn0ioo2  45704  vonn0icc2  45706