Theorem nfsb4t 2024

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2022). (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 1554	. . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑
2	1	nfal 1586	. . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑
3		nfnae 1732	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
4	2, 3	nfan 1575	. . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5		df-nf 1471	. . . . . 6 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
6	5	albii 1480	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧𝜑 ↔ ∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑))
7		hbsb4t 2023	. . . . 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 1533	. 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 1361  Ⅎwnf 1470  [wsb 1772

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773

This theorem is referenced by:  dvelimdf  2026