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Theorem hbsb4t 1993
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1992). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
hbsb4t (∀𝑥𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))

Proof of Theorem hbsb4t
StepHypRef Expression
1 hba1 1520 . . 3 (∀𝑧𝜑 → ∀𝑧𝑧𝜑)
21hbsb4 1992 . 2 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑))
3 spsbim 1823 . . . . 5 (∀𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
43sps 1517 . . . 4 (∀𝑧𝑥(𝜑 → ∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑))
5 ax-4 1490 . . . . . . 7 (∀𝑧𝜑𝜑)
65sbimi 1744 . . . . . 6 ([𝑦 / 𝑥]∀𝑧𝜑 → [𝑦 / 𝑥]𝜑)
76alimi 1435 . . . . 5 (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
87a1i 9 . . . 4 (∀𝑧𝑥(𝜑 → ∀𝑧𝜑) → (∀𝑧[𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
94, 8imim12d 74 . . 3 (∀𝑧𝑥(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
109a7s 1434 . 2 (∀𝑥𝑧(𝜑 → ∀𝑧𝜑) → (([𝑦 / 𝑥]∀𝑧𝜑 → ∀𝑧[𝑦 / 𝑥]∀𝑧𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
112, 10syl5 32 1 (∀𝑥𝑧(𝜑 → ∀𝑧𝜑) → (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1333  [wsb 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by:  nfsb4t  1994
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