Theorem raldifeq 4503

Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)

Hypotheses

Ref	Expression
raldifeq.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
raldifeq.2	⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)

Assertion

Ref	Expression
raldifeq	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem raldifeq

Step	Hyp	Ref	Expression
1		raldifeq.2	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)
2	1	biantrud 531	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)))
3		ralunb 4210	. . 3 ⊢ (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓))
4	2, 3	bitr4di 289	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓))
5		raldifeq.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
6		undif 4491	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
8	7	raleqdv 3326	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
9	4, 8	bitrd 279	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∀wral 3061   ∖ cdif 3963   ∪ cun 3964   ⊆ wss 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343

This theorem is referenced by:  cantnfrescl  9723  rrxmet  25467  ntrneiel2  44092  ntrneik4w  44106