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Theorem sbnf2 2533
Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.)
Assertion
Ref Expression
sbnf2 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbnf2
StepHypRef Expression
1 nfv 2009 . . . . . 6 𝑦𝜑
21sb8e 2516 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 nfv 2009 . . . . . 6 𝑧𝜑
43sb8 2515 . . . . 5 (∀𝑥𝜑 ↔ ∀𝑧[𝑧 / 𝑥]𝜑)
52, 4imbi12i 341 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
6 df-nf 1879 . . . 4 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7 pm11.53v 2039 . . . 4 (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
85, 6, 73bitr4i 294 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
93sb8e 2516 . . . . . 6 (∃𝑥𝜑 ↔ ∃𝑧[𝑧 / 𝑥]𝜑)
101sb8 2515 . . . . . 6 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
119, 10imbi12i 341 . . . . 5 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
12 pm11.53v 2039 . . . . 5 (∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
1311, 12bitr4i 269 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ ∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
14 alcom 2201 . . . 4 (∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
156, 13, 143bitri 288 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
168, 15anbi12i 620 . 2 ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
17 pm4.24 559 . 2 (Ⅎ𝑥𝜑 ↔ (Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑))
18 2albiim 1988 . 2 (∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
1916, 17, 183bitr4i 294 1 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  wnf 1878  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-sb 2063
This theorem is referenced by:  sbnfc2  4171  nfnid  5013
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