Theorem ralbi12f 35430

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 3165	. 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 3396	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	sylan9bbr 513	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531  Ⅎwnfc 2959  ∀wral 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141

This theorem is referenced by: (None)