Theorem nfsb4t 2067

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2065). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)

Assertion

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

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		nfnf1 1593	. . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑
2	1	nfal 1625	. . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑
3		nfnae 1770	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
4	2, 3	nfan 1614	. . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5		df-nf 1510	. . . . . 6 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
6	5	albii 1519	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 ↔ ∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑))
7		hbsb4t 2066	. . . . 5 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
8	6, 7	sylbi 121	. . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
9	8	imp 124	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
10	4, 9	nfd 1572	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
11	10	ex 115	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1396  Ⅎwnf 1509  [wsb 1810

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811

This theorem is referenced by:  dvelimdf  2069