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Theorem raleqbii 2973
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 2680 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3imbi12i 339 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 2961 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ral 2901
This theorem is referenced by:  wfrlem5  7284  ply1coe  19436  ordtbaslem  20750  iscusp2  21864  frrlem5  30822  elghomOLD  32650  iscrngo2  32760  tendoset  34859  comptiunov2i  36811  isrgr  40751
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