Theorem ralbi12f 34989

Description: Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
ralbi12f.1	⊢ Ⅎ𝑥𝐴
ralbi12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
ralbi12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Proof of Theorem ralbi12f

Step	Hyp	Ref	Expression
1		ralbi 3134	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))
2		ralbi12f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		ralbi12f.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	raleqf 3357	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	sylan9bbr 511	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  Ⅎwnfc 2933  ∀wral 3105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110

This theorem is referenced by: (None)