Theorem raleqd 42575

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 3323	. 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 205   = wceq 1539  Ⅎwnfc 2886  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068

This theorem is referenced by:  allbutfiinf  42850