Theorem bj-equsalhv 37482

Description: Version of equsalh 2455 with a disjoint variable condition, which does not require ax-13 2407. Remark: this is the same as equsalhw 2329. TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw 2038 has been moved to Main; Theorem ax13lem2 2411 has a DV version which is a simple consequence of ax5e 1945; Theorems nfeqf2 2412, dveeq2 2413, nfeqf1 2414, dveeq1 2415, nfeqf 2416, axc9 2417, ax13 2410, have dv versions which are simple consequences of ax-5 1943. (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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-equsalhv	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)

Proof of Theorem bj-equsalhv

Step	Hyp	Ref	Expression
1		bj-equsalhv.nf	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2184	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-equsalhv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2306	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)