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Theorem sbnf2 2568
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 1984 . . . . . 6 𝑦𝜑
21sb8e 2554 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 nfv 1984 . . . . . 6 𝑧𝜑
43sb8 2553 . . . . 5 (∀𝑥𝜑 ↔ ∀𝑧[𝑧 / 𝑥]𝜑)
52, 4imbi12i 339 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
6 df-nf 1851 . . . 4 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7 pm11.53v 2063 . . . 4 (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∀𝑧[𝑧 / 𝑥]𝜑))
85, 6, 73bitr4i 292 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
93sb8e 2554 . . . . . 6 (∃𝑥𝜑 ↔ ∃𝑧[𝑧 / 𝑥]𝜑)
101sb8 2553 . . . . . 6 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
119, 10imbi12i 339 . . . . 5 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
12 pm11.53v 2063 . . . . 5 (∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ (∃𝑧[𝑧 / 𝑥]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
1311, 12bitr4i 267 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ ∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
14 alcom 2178 . . . 4 (∀𝑧𝑦([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
156, 13, 143bitri 286 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
168, 15anbi12i 735 . 2 ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
17 pm4.24 678 . 2 (Ⅎ𝑥𝜑 ↔ (Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑))
18 2albiim 1958 . 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 292 1 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1622  wex 1845  wnf 1849  [wsb 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1846  df-nf 1851  df-sb 2039
This theorem is referenced by:  sbnfc2  4142  nfnid  5038
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