Theorem raleqbii 3317

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 2820	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		raleqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	imbi12i 350	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))
5	4	ralbii2 3071	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-clel 2803  df-ral 3045

This theorem is referenced by:  fprlem1  8279  frrlem15  9710  opprdomnb  20626  ply1coe  22185  ordtbaslem  23075  iscusp2  24189  isrgr  29487  iineq12i  36185  elghomOLD  37881  iscrngo2  37991  tendoset  40753  comptiunov2i  43695