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Theorem raleqbii 2644
 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 (x A ψx B χ)

Proof of Theorem raleqbii
StepHypRef Expression
1 raleqbii.1 . . . 4 A = B
21eleq2i 2417 . . 3 (x Ax B)
3 raleqbii.2 . . 3 (ψχ)
42, 3imbi12i 316 . 2 ((x Aψ) ↔ (x Bχ))
54ralbii2 2642 1 (x A ψx B χ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ral 2619 This theorem is referenced by: (None)
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