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Theorem raleqbii 2379
 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 (𝜓𝜒)
Assertion
Ref Expression
raleqbii (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)

Proof of Theorem raleqbii
StepHypRef Expression
1 raleqbii.1 . . . 4 𝐴 = 𝐵
21eleq2i 2146 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3imbi12i 237 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 2377 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
 Colors of variables: wff set class Syntax hints:   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∀wral 2349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-ral 2354 This theorem is referenced by: (None)
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