Theorem bj-equsalhv 36789

Description: Version of equsalh 2419 with a disjoint variable condition, which does not require ax-13 2371. Remark: this is the same as equsalhw 2291. TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw 2005 has been moved to Main; Theorem ax13lem2 2375 has a DV version which is a simple consequence of ax5e 1912; Theorems nfeqf2 2376, dveeq2 2377, nfeqf1 2378, dveeq1 2379, nfeqf 2380, axc9 2381, ax13 2374, have dv versions which are simple consequences of ax-5 1910. (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 2147	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-equsalhv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2268	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)