ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2rexbiia GIF version

Theorem 2rexbiia 2486
Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
2rexbiia.1 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
Assertion
Ref Expression
2rexbiia (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbiia
StepHypRef Expression
1 2rexbiia.1 . . 3 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
21rexbidva 2467 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜓))
32rexbiia 2485 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2141  wrex 2449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-rex 2454
This theorem is referenced by:  elq  9568  cnref1o  9596
  Copyright terms: Public domain W3C validator