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

Theorem rexeqbii 3323
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 2901 . . 3 (𝑥𝐴𝑥𝐵)
3 rexeqbii.2 . . 3 (𝜓𝜒)
42, 3anbi12i 626 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54rexbii2 3242 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-rex 3141
This theorem is referenced by:  bnj882  32097  satfbrsuc  32510
  Copyright terms: Public domain W3C validator