Theorem hbsb4t 2040

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2039). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
hbsb4t	⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))

Proof of Theorem hbsb4t

Step	Hyp	Ref	Expression
1		hba1 1562	. . 3 ⊢ (∀𝑧𝜑 → ∀𝑧∀𝑧𝜑)
2	1	hbsb4 2039	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑))
3		spsbim 1865	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
4	3	sps 1559	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
5		ax-4 1532	. . . . . . 7 ⊢ (∀𝑧𝜑 → 𝜑)
6	5	sbimi 1786	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 → [𝑦 / 𝑥]𝜑)
7	6	alimi 1477	. . . . 5 ⊢ (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
8	7	a1i 9	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
9	4, 8	imim12d 74	. . 3 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
10	9	a7s 1476	. 2 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
11	2, 10	syl5 32	1 ⊢ (∀𝑥∀𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1370  [wsb 1784

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-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by:  nfsb4t  2041