Theorem ixpeq2dva 8850

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 3128	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8849	. 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 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Xcixp 8835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-ss 3918  df-ixp 8836

This theorem is referenced by:  ixpeq2dv  8851  dfac9  10047  xpsrnbas  17492  funcpropd  17826  natpropd  17903  prdsmgp  20086  frlmip  21733  elptr2  23518  dfac14  23562  xkoptsub  23598  prdsxmslem2  24473  rrxip  25346  ptrest  37820  prdsbnd2  37996  hoidmvlelem3  46841  ovnhoilem1  46845  ovnhoilem2  46846  hoicoto2  46849  ovnlecvr2  46854  ovncvr2  46855  ovnovollem1  46900  ovnovollem2  46901  hoimbl2  46909  vonhoire  46916  iccvonmbllem  46922  vonioolem2  46925  vonicclem2  46928  vonn0ioo2  46934  vonn0icc2  46936