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Theorem equsalh 2280
 Description: An equivalence related to implicit substitution. (Contributed by NM, 2-Jun-1993.)
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.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2011 . 2 𝑥𝜓
3 equsalh.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsal 2279 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  dvelimf-o  33232
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