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Theorem bj-equsalhv 34524
 Description: Version of equsalh 2431 with a disjoint variable condition, which does not require ax-13 2379. Remark: this is the same as equsalhw 2295. TODO: delete after moving the following paragraph somewhere. Remarks: equsexvw 2011 has been moved to Main; Theorem ax13lem2 2383 has a DV version which is a simple consequence of ax5e 1913; Theorems nfeqf2 2384, dveeq2 2385, nfeqf1 2386, dveeq1 2387, nfeqf 2388, axc9 2389, ax13 2382, have dv versions which are simple consequences of ax-5 1911. (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 2147 . 2 𝑥𝜓
3 bj-equsalhv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsalv 2265 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
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