Theorem sbnf2 2000

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 1502	. 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 1475	. . . . 5 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3		sbhb 1959	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
4	3	albii 1484	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
5		alcom 1492	. . . . 5 ⊢ (∀𝑥∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
6	2, 4, 5	3bitri 206	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
7		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → [𝑧 / 𝑥]𝜑)
8	7	sb8 1870	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑))
9		nfs1v 1958	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
10	9	sblim 1976	. . . . . . 7 ⊢ ([𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
11	10	albii 1484	. . . . . 6 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
12	8, 11	bitri 184	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
13	12	albii 1484	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
14		alcom 1492	. . . 4 ⊢ (∀𝑧∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
15	6, 13, 14	3bitri 206	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
16		sbhb 1959	. . . . . 6 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
17	16	albii 1484	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
18		alcom 1492	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
19	2, 17, 18	3bitri 206	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
20		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 → [𝑦 / 𝑥]𝜑)
21	20	sb8 1870	. . . . . 6 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑))
22		nfs1v 1958	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
23	22	sblim 1976	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
24	23	albii 1484	. . . . . 6 ⊢ (∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
25	21, 24	bitri 184	. . . . 5 ⊢ (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
26	25	albii 1484	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
27	19, 26	bitri 184	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
28	15, 27	anbi12i 460	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
29		anidm 396	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ Ⅎ𝑥𝜑)
30	1, 28, 29	3bitr2ri 209	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362  Ⅎwnf 1474  [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  sbnfc2  3145