Theorem nfbidv

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

	Ref	Expression
Hypothesis	albidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	nfbidv	⊢ ( 𝜑 → ( Ⅎ 𝑥 𝜓 ↔ Ⅎ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		albidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	exbidv	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )
3	1	albidv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑥 𝜒 ) )
4	2 3	imbi12d	⊢ ( 𝜑 → ( ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜒 → ∀ 𝑥 𝜒 ) ) )
5		df-nf	⊢ ( Ⅎ 𝑥 𝜓 ↔ ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) )
6		df-nf	⊢ ( Ⅎ 𝑥 𝜒 ↔ ( ∃ 𝑥 𝜒 → ∀ 𝑥 𝜒 ) )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( Ⅎ 𝑥 𝜓 ↔ Ⅎ 𝑥 𝜒 ) )