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Theorem bj-equsalhv 33595
 Description: Version of equsalh 2355 with a disjoint variable condition, which does not require ax-13 2301. Remark: this is the same as equsalhw 2225 (TODO: delete after moving the following paragraph somewhere). Remarks: equsexvw 1962 has been moved to Main; the theorem ax13lem2 2305 has a dv version which is a simple consequence of ax5e 1871; the theorems nfeqf2 2306, dveeq2 2308, nfeqf1 2309, dveeq1 2310, nfeqf 2311, axc9 2312, ax13 2304, have dv versions which are simple consequences of ax-5 1869. (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 2084 . 2 𝑥𝜓
3 bj-equsalhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsalv 2196 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106 This theorem depends on definitions:  df-bi 199  df-ex 1743  df-nf 1747 This theorem is referenced by: (None)
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