Theorem bj-nfbiit 32024

Description: Closed form of nfbii 1770 (the label " nfbi 1821 " is taken for another result. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nfbiit

Step	Hyp	Ref	Expression
1		exbi 1762	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2		albi 1736	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
3	1, 2	imbi12d 333	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
4		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
6	3, 4, 5	3bitr4g 302	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)