Theorem raleqbii 2489

Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
raleqbii.1	⊢ 𝐴 = 𝐵
raleqbii.2	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
raleqbii	⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)

Proof of Theorem raleqbii

Step	Hyp	Ref	Expression
1		raleqbii.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2244	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		raleqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	imbi12i 239	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))
5	4	ralbii2 2487	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-ral 2460

This theorem is referenced by: (None)