Theorem drnf1v 2406

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2478 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2179. (Revised by GG, 18-Nov-2024.)

Hypothesis

Ref	Expression
dral1v.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
drnf1v	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem drnf1v

Step	Hyp	Ref	Expression
1		dral1v.1	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	drex1v 2405	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
3	1	dral1v 2404	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
4	2, 3	imbi12d 347	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦𝜓 → ∀𝑦𝜓)))
5		df-nf 1817	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6		df-nf 1817	. 2 ⊢ (Ⅎ𝑦𝜓 ↔ (∃𝑦𝜓 → ∀𝑦𝜓))
7	4, 5, 6	3bitr4g 317	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

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-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  nfriotadw  7388