Theorem raldifeq 4492

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 530	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)))
3		ralunb 4190	. . 3 ⊢ (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓))
4	2, 3	bitr4di 288	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓))
5		raldifeq.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
6		undif 4480	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
7	5, 6	sylib 217	. . 3 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
8	7	raleqdv 3323	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
9	4, 8	bitrd 278	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539  ∀wral 3059   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322

This theorem is referenced by:  cantnfrescl  9673  rrxmet  25156  ntrneiel2  43139  ntrneik4w  43153