Theorem nfbii 1770

Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.)

Hypothesis

Ref	Expression
nfbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
nfbii	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbii

Step	Hyp	Ref	Expression
1		nfbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	exbii 1764	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)
3	1	albii 1737	. . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	2, 3	imbi12i 339	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
5		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7	4, 5, 6	3bitr4i 291	1 ⊢ (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:  nfxfr  1771  nfxfrd  1772  dvelimhw  2159  nfeqf1  2287  dfnfc2  4390  dfnfc2OLD  4391  bj-dvelimdv1  32028  bj-nfcf  32112  iunconlem2  38193