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Theorem equsexhv 2227
 Description: Version of equsexh 2357 with a disjoint variable condition, which does not require ax-13 2302. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsalhw.1 (𝜓 → ∀𝑥𝜓)
equsalhw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexhv (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexhv
StepHypRef Expression
1 equsalhw.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2085 . 2 𝑥𝜓
3 equsalhw.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsexv 2198 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1506  ∃wex 1743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-nf 1748 This theorem is referenced by:  cleljustALT  2297
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