Theorem raldifeq 4494

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 533	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)))
3		ralunb 4192	. . 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 4482	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
7	5, 6	sylib 217	. . 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 205   ∧ wa 397   = wceq 1542  ∀wral 3062   ∖ cdif 3946   ∪ cun 3947   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324

This theorem is referenced by:  cantnfrescl  9671  rrxmet  24925  ntrneiel2  42837  ntrneik4w  42851