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Theorem equsalh 2291
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 2021 . 2 𝑥𝜓
3 equsalh.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsal 2290 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  dvelimf-o  33733
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