Theorem iineq1d 44351

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 5007	. 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 1533  ∩ ciin 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-ral 3056  df-rex 3065  df-iin 4993

This theorem is referenced by:  iineq12dv  44367  smflimlem2  46057  smflimlem3  46058  smflimlem4  46059  smflim2  46091  smflimsuplem1  46105  smflimsuplem7  46111  smflimsup  46113  smfliminf  46116