MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  raleqbii Structured version   Visualization version   GIF version

Theorem raleqbii 3128
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 339 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54ralbii2 3116 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756  df-ral 3055
This theorem is referenced by:  wfrlem5  7589  ply1coe  19888  ordtbaslem  21214  iscusp2  22327  isrgr  26686  frrlem5  32111  elghomOLD  34017  iscrngo2  34127  tendoset  36567  comptiunov2i  38518
  Copyright terms: Public domain W3C validator