Theorem iineq1d 45868

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 4976	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 18	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∩ ciin 4959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-ral 3082  df-rex 3092  df-iin 4961

This theorem is used by:  smflimlem2  47546  smflimlem3  47547  smflimlem4  47548  smflim2  47580  smflimsuplem1  47594  smflimsuplem7  47600  smflimsup  47602  smfliminf  47605