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Theorem bj-equsalhv 34026
Description: Version of equsalh 2436 with a disjoint variable condition, which does not require ax-13 2383. Remark: this is the same as equsalhw 2291. TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw 2002 has been moved to Main; the theorem ax13lem2 2387 has a dv version which is a simple consequence of ax5e 1904; the theorems nfeqf2 2388, dveeq2 2389, nfeqf1 2390, dveeq1 2391, nfeqf 2392, axc9 2393, ax13 2386, have dv versions which are simple consequences of ax-5 1902. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage 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 2141 . 2 𝑥𝜓
3 bj-equsalhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsalv 2259 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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