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Theorem equsexhv 2146
Description: Version of equsexh 2294 with a dv condition, which does not require ax-13 2245. (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 2021 . 2 𝑥𝜓
3 equsexhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsexv 2106 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  cleljustALT  2184
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