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Theorem cbvexd 1845
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1936. (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 1453 . 2 (𝜑 → ∀𝑦𝜑)
3 cbvexd.2 . . 3 (𝜑 → Ⅎ𝑦𝜓)
43nfrd 1454 . 2 (𝜑 → (𝜓 → ∀𝑦𝜓))
5 cbvexd.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
62, 4, 5cbvexdh 1844 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbvexdva  1847  vtoclgft  2660  bdsepnft  11121  strcollnft  11222
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