Theorem raleqd 42300

Description: Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
raleqd.a	⊢ Ⅎ𝑥𝐴
raleqd.b	⊢ Ⅎ𝑥𝐵
raleqd.e	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Proof of Theorem raleqd

Step	Hyp	Ref	Expression
1		raleqd.e	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqd.a	. . 3 ⊢ Ⅎ𝑥𝐴
3		raleqd.b	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	raleqf 3299	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	syl 17	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543  Ⅎwnfc 2877  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056

This theorem is referenced by:  allbutfiinf  42574