Theorem raleqbid 3323

Description: Equality deduction for restricted universal quantifier. See raleqbidv 3314 for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
raleqbid.0	⊢ Ⅎ𝑥𝜑
raleqbid.1	⊢ Ⅎ𝑥𝐴
raleqbid.2	⊢ Ⅎ𝑥𝐵
raleqbid.3	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbid.4	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
raleqbid	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Proof of Theorem raleqbid

Step	Hyp	Ref	Expression
1		raleqbid.3	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbid.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		raleqbid.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	raleqf 3321	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
6		raleqbid.0	. . 3 ⊢ Ⅎ𝑥𝜑
7		raleqbid.4	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
8	6, 7	ralbid 3253	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
9	5, 8	bitrd 280	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  Ⅎwnf 1790  Ⅎwnfc 2887  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055

This theorem is referenced by: (None)