Theorem bj-equsalhv 37810

Description: Version of equsalh 2449 with a disjoint variable condition, which does not require ax-13 2401. Remark: this is the same as equsalhw 2324. TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw 2037 has been moved to Main; Theorem ax13lem2 2405 has a DV version which is a simple consequence of ax5e 1944; Theorems nfeqf2 2406, dveeq2 2407, nfeqf1 2408, dveeq1 2409, nfeqf 2410, axc9 2411, ax13 2404, have dv versions which are simple consequences of ax-5 1942. (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 2182	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-equsalhv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2301	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by: (None)