Theorem hbsb4t 2042

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2041). (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 1564	. . 3 ⊢ (∀𝑧𝜑 → ∀𝑧∀𝑧𝜑)
2	1	hbsb4 2041	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑))
3		spsbim 1867	. . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
4	3	sps 1561	. . . 4 ⊢ (∀𝑧∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
5		ax-4 1534	. . . . . . 7 ⊢ (∀𝑧𝜑 → 𝜑)
6	5	sbimi 1788	. . . . . 6 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 → [𝑦 / 𝑥]𝜑)
7	6	alimi 1479	. . . . 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 1478	. 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 1371  [wsb 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  nfsb4t  2043