Theorem ixpeq1d 8940

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 8939	. 2 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 18	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Xcixp 8928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-fn 6541  df-ixp 8929

This theorem is used by:  elixpsn  8968  ixpsnf1o  8969  dfac9  10218  prdsval  17631  isfunc  18044  funcpropd  18082  natfval  18129  natpropd  18159  dprdval  20224  ptval  23894  dfac14  23942  ptuncnv  24131  ptunhmeo  24132  hoidmvle  47703  hoimbl  47734