Theorem ixpeq2dva 8933

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 3155	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8932	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
4	2, 3	syl 18	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Xcixp 8918

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-ss 3916  df-ixp 8919

This theorem is used by:  ixpeq2dv  8934  dfac9  10208  xpsrnbas  17736  funcpropd  18070  natpropd  18147  prdsmgp  20364  frlmip  22077  elptr2  23886  dfac14  23930  xkoptsub  23966  prdsxmslem2  24841  rrxip  25704  ptrest  38517  prdsbnd2  38709  hoidmvlelem3  47576  ovnhoilem1  47580  ovnhoilem2  47581  hoicoto2  47584  ovnlecvr2  47589  ovncvr2  47590  ovnovollem1  47635  ovnovollem2  47636  hoimbl2  47644  vonhoire  47651  iccvonmbllem  47657  vonioolem2  47660  vonicclem2  47663  vonn0ioo2  47669  vonn0icc2  47671