Theorem nfbidf 2261

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1806 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

Hypotheses

Ref	Expression
albid.1	⊢ Ⅎ𝑥𝜑
albid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
nfbidf	⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Proof of Theorem nfbidf

Step	Hyp	Ref	Expression
1		albid.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		albid.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	exbid 2260	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 2	albid 2259	. . 3 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
5	3, 4	imbi12d 346	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ (∃𝑥𝜒 → ∀𝑥𝜒)))
6		df-nf 1806	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7		df-nf 1806	. 2 ⊢ (Ⅎ𝑥𝜒 ↔ (∃𝑥𝜒 → ∀𝑥𝜒))
8	5, 6, 7	3bitr4g 316	1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560  ∃wex 1801  Ⅎwnf 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-ex 1802  df-nf 1806

This theorem is referenced by:  drnf2  2477  dvelimdf  2482  nfceqdf  2922  wl-nfimf1  38034  ichnfimlem  48074