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

Theorem cbvexd 1818
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1909. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
cbvexd.1 𝑦𝜑
cbvexd.2 (𝜑 → Ⅎ𝑦𝜓)
cbvexd.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvexd (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvexd.1 . . 3 𝑦𝜑
21nfri 1428 . 2 (𝜑 → ∀𝑦𝜑)
3 cbvexd.2 . . 3 (𝜑 → Ⅎ𝑦𝜓)
43nfrd 1429 . 2 (𝜑 → (𝜓 → ∀𝑦𝜓))
5 cbvexd.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
62, 4, 5cbvexdh 1817 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wnf 1365  wex 1397
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
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  cbvexdva  1820  vtoclgft  2621  bdsepnft  10366  strcollnft  10468
  Copyright terms: Public domain W3C validator