Theorem ixpeq2dva 8854

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 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8853	. 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 1542   ∈ wcel 2114  ∀wral 3052  Xcixp 8839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-ss 3907  df-ixp 8840

This theorem is referenced by:  ixpeq2dv  8855  dfac9  10053  xpsrnbas  17529  funcpropd  17863  natpropd  17940  prdsmgp  20126  frlmip  21771  elptr2  23552  dfac14  23596  xkoptsub  23632  prdsxmslem2  24507  rrxip  25370  ptrest  37957  prdsbnd2  38133  hoidmvlelem3  47046  ovnhoilem1  47050  ovnhoilem2  47051  hoicoto2  47054  ovnlecvr2  47059  ovncvr2  47060  ovnovollem1  47105  ovnovollem2  47106  hoimbl2  47114  vonhoire  47121  iccvonmbllem  47127  vonioolem2  47130  vonicclem2  47133  vonn0ioo2  47139  vonn0icc2  47141