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

Theorem cbvrexdva 2557
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvraldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvrexdva (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvrexdva
StepHypRef Expression
1 cbvraldva.1 . 2 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
2 eqidd 2057 . 2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐴)
31, 2cbvrexdva2 2555 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-rex 2329
This theorem is referenced by:  tfrlem3ag  5955  tfrlem3a  5956  tfrlemi1  5977
  Copyright terms: Public domain W3C validator