Theorem iineq1d 45276

Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypothesis

Ref	Expression
iineq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
iineq1d	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem iineq1d

Step	Hyp	Ref	Expression
1		iineq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		iineq1 4962	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1541  ∩ ciin 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-ral 3050  df-rex 3059  df-iin 4947

This theorem is referenced by:  smflimlem2  46958  smflimlem3  46959  smflimlem4  46960  smflim2  46992  smflimsuplem1  47006  smflimsuplem7  47012  smflimsup  47014  smfliminf  47017