Theorem raleqbii 3316

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 2829	. . 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 3080	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812  df-ral 3053

This theorem is referenced by:  fprlem1  8252  frrlem15  9681  opprdomnb  20662  ply1coe  22254  ordtbaslem  23144  iscusp2  24257  isrgr  29645  iineq12i  36413  elghomOLD  38138  iscrngo2  38248  tendoset  41135  comptiunov2i  44062