Theorem iineq12dv 42609

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
iineq12dv.1	⊢ (𝜑 → 𝐴 = 𝐵)
iineq12dv.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
iineq12dv	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

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

Proof of Theorem iineq12dv

Step	Hyp	Ref	Expression
1		iineq12dv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	iineq1d 42593	. 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
3		iineq12dv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
4	3	iineq2dv 4954	. 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐵 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
5	2, 4	eqtrd 2779	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∩ ciin 4930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-iin 4932

This theorem is referenced by:  smflim  44263