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

Theorem rexeqbii 3202
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
rexeqbii.1 𝐴 = 𝐵
rexeqbii.2 (𝜓𝜒)
Assertion
Ref Expression
rexeqbii (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)

Proof of Theorem rexeqbii
StepHypRef Expression
1 rexeqbii.1 . . . 4 𝐴 = 𝐵
21eleq2i 2842 . . 3 (𝑥𝐴𝑥𝐵)
3 rexeqbii.2 . . 3 (𝜓𝜒)
42, 3anbi12i 612 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54rexbii2 3187 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-clel 2767  df-rex 3067
This theorem is referenced by:  bnj882  31334
  Copyright terms: Public domain W3C validator