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Theorem equsalh 1614
 Description: A useful equivalence related to substitution. New proofs should use equsal 1615 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 1445 . . . . 5 (∀𝑥𝜓𝜓)
41, 3syl6bbr 187 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
54pm5.74i 169 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑥𝜓))
65albii 1359 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
72a1d 22 . . . 4 (𝜓 → (𝑥 = 𝑦 → ∀𝑥𝜓))
82, 7alrimih 1358 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
9 ax9o 1588 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
108, 9impbii 117 . 2 (𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
116, 10bitr4i 176 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  sb6x  1662  dvelimfALT2  1698  dvelimALT  1886  dvelimfv  1887  dvelimor  1894
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