Theorem nfbidv 1925

Description: An equality theorem for nonfreeness. See nfbidf 2217 for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) Remove dependency on ax-6 1971, ax-7 2011, ax-12 2171 by adapting proof of nfbidf 2217. (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 1924	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
3	1	albidv 1923	. . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
4	2, 3	imbi12d 345	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
5		df-nf 1787	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6		df-nf 1787	. 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by:  nfcjust  2888  nfcr  2892  nfcriOLD  2897  nfcriOLDOLD  2898  bj-drnf2v  34992