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 → (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 2084	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-equsalhv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2196	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

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)