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Theorem equsexh 2399
 Description: An equivalence related to implicit substitution. See equsexhv 2266 for a version with a disjoint variable condition which does not require ax-13 2344. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
equsalh.1 (𝜓 → ∀𝑥𝜓)
equsalh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexh (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsexh
StepHypRef Expression
1 equsalh.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2117 . 2 𝑥𝜓
3 equsalh.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsex 2396 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-nf 1766 This theorem is referenced by: (None)
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