Theorem iineq12f 38873

Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
iineq12f.1	⊢ Ⅎ𝑥𝐴
iineq12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
iineq12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iineq12f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2854	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
2	1	ralimi 3104	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		ralbi 3122	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷))
4	2, 3	syl 18	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷))
5		iineq12f.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6		iineq12f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	5, 6	raleqf 3347	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
8	4, 7	sylan9bbr 520	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
9	8	abbidv 2831	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
10		df-iin 4961	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
11		df-iin 4961	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
12	9, 10, 11	3eqtr4g 2825	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912  ∀wral 3081  ∩ 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-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-iin 4961

This theorem is used by: (None)