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Theorem equsexhv 2135
 Description: Version of equsexh 2283 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsexhv.nf (𝜓 → ∀𝑥𝜓)
equsexhv.1 (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 equsexhv.nf . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2011 . 2 𝑥𝜓
3 equsexhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsexv 2095 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 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-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  cleljustALT  2173
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