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Theorem raleqbii 3172
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 2871 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3imbi12i 342 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 3160 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  wral 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2793  df-clel 2796  df-ral 3095
This theorem is referenced by:  wfrlem5  7659  ply1coe  19987  ordtbaslem  21320  iscusp2  22433  isrgr  26808  frrlem5  32296  elghomOLD  34172  iscrngo2  34282  tendoset  36779  comptiunov2i  38776
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