Theorem ixpeq1d 8905

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 8904	. 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 1539  Xcixp 8893

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-fn 6545  df-ixp 8894

This theorem is referenced by:  elixpsn  8933  ixpsnf1o  8934  dfac9  10133  prdsval  17405  isfunc  17818  funcpropd  17855  natfval  17901  natpropd  17933  dprdval  19914  ptval  23294  dfac14  23342  ptuncnv  23531  ptunhmeo  23532  hoidmvle  45614  hoimbl  45645