Theorem nfsb4t 2080

Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2081). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)

Assertion

Ref	Expression
nfsb4t	⊢ (∀xℲzφ → (¬ ∀z z = y → Ⅎz[y / x]φ))

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		sbequ12 1919	. . . . . . . . 9 ⊢ (x = y → (φ ↔ [y / x]φ))
2	1	sps 1754	. . . . . . . 8 ⊢ (∀x x = y → (φ ↔ [y / x]φ))
3	2	drnf2 1970	. . . . . . 7 ⊢ (∀x x = y → (Ⅎzφ ↔ Ⅎz[y / x]φ))
4	3	biimpcd 215	. . . . . 6 ⊢ (Ⅎzφ → (∀x x = y → Ⅎz[y / x]φ))
5	4	sps 1754	. . . . 5 ⊢ (∀xℲzφ → (∀x x = y → Ⅎz[y / x]φ))
6	5	a1dd 42	. . . 4 ⊢ (∀xℲzφ → (∀x x = y → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz[y / x]φ)))
7		nfa1 1788	. . . . . . . 8 ⊢ Ⅎx∀xℲzφ
8		nfnae 1956	. . . . . . . . 9 ⊢ Ⅎx ¬ ∀z z = x
9		nfnae 1956	. . . . . . . . 9 ⊢ Ⅎx ¬ ∀z z = y
10	8, 9	nfan 1824	. . . . . . . 8 ⊢ Ⅎx(¬ ∀z z = x ∧ ¬ ∀z z = y)
11	7, 10	nfan 1824	. . . . . . 7 ⊢ Ⅎx(∀xℲzφ ∧ (¬ ∀z z = x ∧ ¬ ∀z z = y))
12		nfeqf 1958	. . . . . . . . 9 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz x = y)
13	12	adantl 452	. . . . . . . 8 ⊢ ((∀xℲzφ ∧ (¬ ∀z z = x ∧ ¬ ∀z z = y)) → Ⅎz x = y)
14		sp 1747	. . . . . . . . 9 ⊢ (∀xℲzφ → Ⅎzφ)
15	14	adantr 451	. . . . . . . 8 ⊢ ((∀xℲzφ ∧ (¬ ∀z z = x ∧ ¬ ∀z z = y)) → Ⅎzφ)
16	13, 15	nfimd 1808	. . . . . . 7 ⊢ ((∀xℲzφ ∧ (¬ ∀z z = x ∧ ¬ ∀z z = y)) → Ⅎz(x = y → φ))
17	11, 16	nfald 1852	. . . . . 6 ⊢ ((∀xℲzφ ∧ (¬ ∀z z = x ∧ ¬ ∀z z = y)) → Ⅎz∀x(x = y → φ))
18	17	ex 423	. . . . 5 ⊢ (∀xℲzφ → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz∀x(x = y → φ)))
19		nfnae 1956	. . . . . . 7 ⊢ Ⅎz ¬ ∀x x = y
20		sb4b 2054	. . . . . . 7 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))
21	19, 20	nfbidf 1774	. . . . . 6 ⊢ (¬ ∀x x = y → (Ⅎz[y / x]φ ↔ Ⅎz∀x(x = y → φ)))
22	21	imbi2d 307	. . . . 5 ⊢ (¬ ∀x x = y → (((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz[y / x]φ) ↔ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz∀x(x = y → φ))))
23	18, 22	syl5ibrcom 213	. . . 4 ⊢ (∀xℲzφ → (¬ ∀x x = y → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz[y / x]φ)))
24	6, 23	pm2.61d 150	. . 3 ⊢ (∀xℲzφ → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz[y / x]φ))
25	24	exp3a 425	. 2 ⊢ (∀xℲzφ → (¬ ∀z z = x → (¬ ∀z z = y → Ⅎz[y / x]φ)))
26		nfsb2 2058	. . 3 ⊢ (¬ ∀z z = y → Ⅎz[y / z]φ)
27		drsb1 2022	. . . 4 ⊢ (∀z z = x → ([y / z]φ ↔ [y / x]φ))
28	27	drnf2 1970	. . 3 ⊢ (∀z z = x → (Ⅎz[y / z]φ ↔ Ⅎz[y / x]φ))
29	26, 28	syl5ib 210	. 2 ⊢ (∀z z = x → (¬ ∀z z = y → Ⅎz[y / x]φ))
30	25, 29	pm2.61d2 152	1 ⊢ (∀xℲzφ → (¬ ∀z z = y → Ⅎz[y / x]φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  nfsb4  2081  dvelimdf  2082  nfsbd  2111