Theorem ralbi12f 38837

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 3120	. 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 3345	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	sylan9bbr 519	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  Ⅎwnfc 2910  ∀wral 3079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080

This theorem is used by: (None)