Theorem nfbiit 1851

Description: Equivalence theorem for the nonfreeness predicate. Closed form of nfbii 1852. (Contributed by Giovanni Mascellani, 10-Apr-2018.) Reduce axiom usage. (Revised by BJ, 6-May-2019.)

Assertion

Ref	Expression
nfbiit	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Proof of Theorem nfbiit

Step	Hyp	Ref	Expression
1		exbi 1847	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2		albi 1818	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
3	1, 2	imbi12d 344	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
4		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6	3, 4, 5	3bitr4g 314	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  nfbii  1852