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Theorem bj-equsalhv 32440
Description: Version of equsalh 2291 with a dv condition, which does not require ax-13 2245. Remark: this is the same as equsalhw 2120.

Remarks: equsexvw 1929 has been moved to Main; the theorem ax13lem2 2295 has a dv version which is a simple consequence of ax5e 1838; the theorems nfeqf2 2296, dveeq2 2297, nfeqf1 2298, dveeq1 2299, nfeqf 2300, axc9 2301, ax13 2248, have dv versions which are simple consequences of ax-5 1836. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Hypotheses
Ref Expression
bj-equsalhv.nf (𝜓 → ∀𝑥𝜓)
bj-equsalhv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-equsalhv (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-equsalhv
StepHypRef Expression
1 bj-equsalhv.nf . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2021 . 2 𝑥𝜓
3 bj-equsalhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3bj-equsalv 32439 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
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-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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