Theorem ixpeq1d 8895

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 8894	. 2 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 18	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1563  Xcixp 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-fn 6528  df-ixp 8884

This theorem is referenced by:  elixpsn  8923  ixpsnf1o  8924  dfac9  10108  prdsval  17498  isfunc  17911  funcpropd  17949  natfval  17996  natpropd  18026  dprdval  20066  ptval  23688  dfac14  23736  ptuncnv  23925  ptunhmeo  23926  hoidmvle  47172  hoimbl  47203