Theorem sbnf2 1898

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.)

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		2albiim 1417	. 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)))
2		df-nf 1390	. . . . 5 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3		sbhb 1857	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
4	3	albii 1399	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
5		alcom 1407	. . . . 5 ⊢ (∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
6	2, 4, 5	3bitri 204	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
7		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → [𝑧 / 𝑥]𝜑)
8	7	sb8 1777	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑))
9		nfs1v 1856	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
10	9	sblim 1872	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
11	10	albii 1399	. . . . . 6 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
12	8, 11	bitri 182	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
13	12	albii 1399	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
14		alcom 1407	. . . 4 ⊢ (∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
15	6, 13, 14	3bitri 204	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
16		sbhb 1857	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
17	16	albii 1399	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
18		alcom 1407	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
19	2, 17, 18	3bitri 204	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
20		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜑)
21	20	sb8 1777	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑))
22		nfs1v 1856	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
23	22	sblim 1872	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
24	23	albii 1399	. . . . . 6 ⊢ (∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
25	21, 24	bitri 182	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
26	25	albii 1399	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
27	19, 26	bitri 182	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
28	15, 27	anbi12i 447	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
29		anidm 388	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ Ⅎ𝑥𝜑)
30	1, 28, 29	3bitr2ri 207	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbnfc2  2962