Theorem raldifeq 4448

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 541	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓)))
3		ralunb 4195	. . 3 ⊢ (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓))
4	2, 3	bitr4di 292	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓))
5		raldifeq.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
6		undif 4437	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
7	5, 6	sylib 221	. . 3 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
8	7	raleqdv 3319	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
9	4, 8	bitrd 282	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045   ∪ cun 4149

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279

This theorem is used by:  cantnfrescl  9677  rrxmet  25729  ntrneiel2  45086  ntrneik4w  45100