Theorem nfbidv 1955

Description: An equality theorem for nonfreeness. See nfbidf 2263 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 2000, ax-7 2041, ax-12 2216 by adapting proof of nfbidf 2263. (Revised by BJ, 25-Sep-2022.)

Hypothesis

Ref	Expression
albidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
nfbidv	⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem nfbidv

Step	Hyp	Ref	Expression
1		albidv.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	exbidv 1954	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
3	1	albidv 1953	. . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
4	2, 3	imbi12d 347	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
5		df-nf 1817	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6		df-nf 1817	. 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

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

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

This theorem is used by:  nfcjust  2913  nfcr  2917  bj-drnf2v  37504