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Theorem raleqbii 3344
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 2833 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3imbi12i 350 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 3089 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816  df-ral 3062
This theorem is referenced by:  fprlem1  8325  wfrlem5OLD  8353  frrlem15  9797  opprdomnb  20717  ply1coe  22302  ordtbaslem  23196  iscusp2  24311  isrgr  29577  iineq12i  36198  elghomOLD  37894  iscrngo2  38004  tendoset  40761  comptiunov2i  43719
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