Theorem nfbidv 1898

Description: An equality theorem for nonfreeness. See nfbidf 2189 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1945, ax-7 1990, ax-12 2139 by adapting proof of nfbidf 2189. (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 1897	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
3	1	albidv 1896	. . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
4	2, 3	imbi12d 346	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
5		df-nf 1764	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6		df-nf 1764	. 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
7	4, 5, 6	3bitr4g 315	1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1518  ∃wex 1759  Ⅎwnf 1763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886

This theorem depends on definitions:  df-bi 208  df-ex 1760  df-nf 1764

This theorem is referenced by:  nfcjust  2932  bj-drnf2v  33626