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Theorem raleqbii 3342
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 2831 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3imbi12i 350 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 3087 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-ral 3060
This theorem is referenced by:  fprlem1  8324  wfrlem5OLD  8352  frrlem15  9795  opprdomnb  20734  ply1coe  22318  ordtbaslem  23212  iscusp2  24327  isrgr  29592  iineq12i  36179  elghomOLD  37874  iscrngo2  37984  tendoset  40742  comptiunov2i  43696
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