Theorem ixpeq1d 8851

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
ixpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ixpeq1d	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq1d

Step	Hyp	Ref	Expression
1		ixpeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ixpeq1 8850	. 2 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1548  Xcixp 8839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-fn 6492  df-ixp 8840

This theorem is referenced by:  elixpsn  8879  ixpsnf1o  8880  dfac9  10054  prdsval  17413  isfunc  17826  funcpropd  17864  natfval  17911  natpropd  17941  dprdval  19975  ptval  23557  dfac14  23605  ptuncnv  23794  ptunhmeo  23795  hoidmvle  47057  hoimbl  47088