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Theorem equsex 2281
 Description: An equivalence related to implicit substitution. See equsexv 2095 for a version with a dv condition which does not require ax-13 2234. See equsexALT 2282 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
Hypotheses
Ref Expression
equsex.nf 𝑥𝜓
equsex.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsex (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsex
StepHypRef Expression
1 equsex.nf . . 3 𝑥𝜓
2 equsex.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpa 500 . . 3 ((𝑥 = 𝑦𝜑) → 𝜓)
41, 3exlimi 2073 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → 𝜓)
51, 2equsal 2279 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
6 equs4 2278 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6sylbir 224 . 2 (𝜓 → ∃𝑥(𝑥 = 𝑦𝜑))
84, 7impbii 198 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 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-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:  equsexh  2283  sb5rf  2410
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