Theorem raleqbii 3336

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 2855	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		raleqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	imbi12i 353	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))
5	4	ralbii2 3107	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∀wral 3079

This theorem was proved from 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-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-clel 2838  df-ral 3080

This theorem is referenced by:  fprlem1  8293  frrlem15  9725  opprdomnb  20815  ply1coe  22458  ordtbaslem  23345  iscusp2  24458  isrgr  29909  iineq12i  36709  elghomOLD  38538  iscrngo2  38648  tendoset  41533  comptiunov2i  44432