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Theorem equsexh 2462
 Description: An equivalence related to implicit substitution. See equsexhv 2300 for a version with a dv condition which does not require ax-13 2420. (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 2190 . 2 𝑥𝜓
3 equsalh.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsex 2459 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384  ∀wal 1635  ∃wex 1859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-10 2185  ax-12 2214  ax-13 2420 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864 This theorem is referenced by: (None)
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