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Theorem sbnf2 1899
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.)
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 2albiim 1418 . 2 (∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
2 df-nf 1391 . . . . 5 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3 sbhb 1858 . . . . . 6 ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
43albii 1400 . . . . 5 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝑧(𝜑 → [𝑧 / 𝑥]𝜑))
5 alcom 1408 . . . . 5 (∀𝑥𝑧(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
62, 4, 53bitri 204 . . . 4 (Ⅎ𝑥𝜑 ↔ ∀𝑧𝑥(𝜑 → [𝑧 / 𝑥]𝜑))
7 nfv 1462 . . . . . . 7 𝑦(𝜑 → [𝑧 / 𝑥]𝜑)
87sb8 1778 . . . . . 6 (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑))
9 nfs1v 1857 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜑
109sblim 1873 . . . . . . 7 ([𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
1110albii 1400 . . . . . 6 (∀𝑦[𝑦 / 𝑥](𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
128, 11bitri 182 . . . . 5 (∀𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
1312albii 1400 . . . 4 (∀𝑧𝑥(𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
14 alcom 1408 . . . 4 (∀𝑧𝑦([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
156, 13, 143bitri 204 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
16 sbhb 1858 . . . . . 6 ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
1716albii 1400 . . . . 5 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
18 alcom 1408 . . . . 5 (∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
192, 17, 183bitri 204 . . . 4 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑥(𝜑 → [𝑦 / 𝑥]𝜑))
20 nfv 1462 . . . . . . 7 𝑧(𝜑 → [𝑦 / 𝑥]𝜑)
2120sb8 1778 . . . . . 6 (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑))
22 nfs1v 1857 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
2322sblim 1873 . . . . . . 7 ([𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
2423albii 1400 . . . . . 6 (∀𝑧[𝑧 / 𝑥](𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
2521, 24bitri 182 . . . . 5 (∀𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
2625albii 1400 . . . 4 (∀𝑦𝑥(𝜑 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
2719, 26bitri 182 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑))
2815, 27anbi12i 448 . 2 ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ (∀𝑦𝑧([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑) ∧ ∀𝑦𝑧([𝑧 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)))
29 anidm 388 . 2 ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜑) ↔ Ⅎ𝑥𝜑)
301, 28, 293bitr2ri 207 1 (Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wnf 1390  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sbnfc2  2963
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