ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexbii2 Structured version   GIF version

Theorem rexbii2 2309
Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
rexbii2.1 ((x A φ) ↔ (x B ψ))
Assertion
Ref Expression
rexbii2 (x A φx B ψ)

Proof of Theorem rexbii2
StepHypRef Expression
1 rexbii2.1 . . 3 ((x A φ) ↔ (x B ψ))
21exbii 1474 . 2 (x(x A φ) ↔ x(x B ψ))
3 df-rex 2286 . 2 (x A φx(x A φ))
4 df-rex 2286 . 2 (x B ψx(x B ψ))
52, 3, 43bitr4i 201 1 (x A φx B ψ)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1358   wcel 1370  wrex 2281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-rex 2286
This theorem is referenced by:  rexeqbii  2311  rexbiia  2313  rexrab  2677  rexdifsn  3469  bnd2  3896
  Copyright terms: Public domain W3C validator