Theorem ixpeq2dva 8888

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 3126	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		ixpeq2 8887	. 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 1540   ∈ wcel 2109  ∀wral 3045  Xcixp 8873

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-ss 3934  df-ixp 8874

This theorem is referenced by:  ixpeq2dv  8889  dfac9  10097  xpsrnbas  17541  funcpropd  17871  natpropd  17948  prdsmgp  20067  frlmip  21694  elptr2  23468  dfac14  23512  xkoptsub  23548  prdsxmslem2  24424  rrxip  25297  ptrest  37620  prdsbnd2  37796  hoidmvlelem3  46602  ovnhoilem1  46606  ovnhoilem2  46607  hoicoto2  46610  ovnlecvr2  46615  ovncvr2  46616  ovnovollem1  46661  ovnovollem2  46662  hoimbl2  46670  vonhoire  46677  iccvonmbllem  46683  vonioolem2  46686  vonicclem2  46689  vonn0ioo2  46695  vonn0icc2  46697