Theorem raleqd 45782

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 3351	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	syl 18	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  Ⅎwnfc 2916  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086

This theorem is referenced by:  allbutfiinf  46061