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Theorem equsexh 2440
 Description: An equivalence related to implicit substitution. See equsexhv 2271 for a version with a dv condition which does not require ax-13 2391. (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 2173 . 2 𝑥𝜓
3 equsalh.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsex 2437 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1630  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859 This theorem is referenced by: (None)
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