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

Theorem equsalh 1726
Description: A useful equivalence related to substitution. New proofs should use equsal 1727 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsalh.1 (𝜓 → ∀𝑥𝜓)
equsalh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsalh (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsalh
StepHypRef Expression
1 equsalh.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
2 equsalh.1 . . . . . 6 (𝜓 → ∀𝑥𝜓)
3219.3h 1553 . . . . 5 (∀𝑥𝜓𝜓)
41, 3bitr4di 198 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
54pm5.74i 180 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑥𝜓))
65albii 1470 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
72a1d 22 . . . 4 (𝜓 → (𝑥 = 𝑦 → ∀𝑥𝜓))
82, 7alrimih 1469 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
9 ax9o 1698 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
108, 9impbii 126 . 2 (𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
116, 10bitr4i 187 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  sb6x  1779  dvelimfALT2  1817  dvelimALT  2010  dvelimfv  2011  dvelimor  2018
  Copyright terms: Public domain W3C validator