Theorem sbnf2 2361

Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) Avoid ax-13 2377. (Revised by Wolf Lammen, 30-Jan-2023.)

Assertion

Ref	Expression
sbnf2	⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbnf2

Step	Hyp	Ref	Expression
1		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑦𝜑
2	1	sb8ef 2358	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3		sb8v 2355	. . . . 5 ⊢ (∀𝑥𝜑 ↔ ∀𝑧[𝑧 / 𝑥]𝜑)
4	2, 3	imbi12i 350	. . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
5		df-nf 1784	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6		pm11.53v 1944	. . . 4 ⊢ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
7	4, 5, 6	3bitr4i 303	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
8		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
9	8	sb8ef 2358	. . . . . 6 ⊢ (∃𝑥𝜑 ↔ ∃𝑧[𝑧 / 𝑥]𝜑)
10		sb8v 2355	. . . . . 6 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
11	9, 10	imbi12i 350	. . . . 5 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
12		pm11.53v 1944	. . . . 5 ⊢ (∀𝑧∀𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
13	11, 5, 12	3bitr4i 303	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑧∀𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
14		alcom 2159	. . . 4 ⊢ (∀𝑧∀𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
15	13, 14	bitri 275	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
16	7, 15	anbi12i 628	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
17		pm4.24 563	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑))
18		2albiim 1890	. 2 ⊢ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
19	16, 17, 18	3bitr4i 303	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by:  sbnfc2  4439  nfnid  5375  ichnfim  47451