Theorem ixpeq2dva 8916

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 3159	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8915	. 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 2146  ∀wral 3081  Xcixp 8901

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-ss 3923  df-ixp 8902

This theorem is used by:  ixpeq2dv  8917  dfac9  10136  xpsrnbas  17647  funcpropd  17981  natpropd  18058  prdsmgp  20271  frlmip  21978  elptr2  23782  dfac14  23826  xkoptsub  23862  prdsxmslem2  24737  rrxip  25600  ptrest  38327  prdsbnd2  38504  hoidmvlelem3  47369  ovnhoilem1  47373  ovnhoilem2  47374  hoicoto2  47377  ovnlecvr2  47382  ovncvr2  47383  ovnovollem1  47428  ovnovollem2  47429  hoimbl2  47437  vonhoire  47444  iccvonmbllem  47450  vonioolem2  47453  vonicclem2  47456  vonn0ioo2  47462  vonn0icc2  47464