Theorem sbnf2 1954

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 1464	. 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 1437	. . . . 5 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3		sbhb 1911	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
4	3	albii 1446	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
5		alcom 1454	. . . . 5 ⊢ (∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
6	2, 4, 5	3bitri 205	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
7		nfv 1508	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → [𝑧 / 𝑥]𝜑)
8	7	sb8 1828	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑))
9		nfs1v 1910	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
10	9	sblim 1928	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
11	10	albii 1446	. . . . . 6 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
12	8, 11	bitri 183	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
13	12	albii 1446	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
14		alcom 1454	. . . 4 ⊢ (∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
15	6, 13, 14	3bitri 205	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
16		sbhb 1911	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
17	16	albii 1446	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
18		alcom 1454	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
19	2, 17, 18	3bitri 205	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
20		nfv 1508	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜑)
21	20	sb8 1828	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑))
22		nfs1v 1910	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
23	22	sblim 1928	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
24	23	albii 1446	. . . . . 6 ⊢ (∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
25	21, 24	bitri 183	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
26	25	albii 1446	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
27	19, 26	bitri 183	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
28	15, 27	anbi12i 455	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
29		anidm 393	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ Ⅎ𝑥𝜑)
30	1, 28, 29	3bitr2ri 208	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbnfc2  3055