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Theorem sbnf2 2108
 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 (Ⅎxφyz([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 1612 . 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 1545 . . . . 5 (Ⅎxφx(φxφ))
3 sbhb 2107 . . . . . 6 ((φxφ) ↔ z(φ → [z / x]φ))
43albii 1566 . . . . 5 (x(φxφ) ↔ xz(φ → [z / x]φ))
5 alcom 1737 . . . . 5 (xz(φ → [z / x]φ) ↔ zx(φ → [z / x]φ))
62, 4, 53bitri 262 . . . 4 (Ⅎxφzx(φ → [z / x]φ))
7 nfv 1619 . . . . . . 7 y(φ → [z / x]φ)
87sb8 2092 . . . . . 6 (x(φ → [z / x]φ) ↔ y[y / x](φ → [z / x]φ))
9 nfs1v 2106 . . . . . . . 8 x[z / x]φ
109sblim 2068 . . . . . . 7 ([y / x](φ → [z / x]φ) ↔ ([y / x]φ → [z / x]φ))
1110albii 1566 . . . . . 6 (y[y / x](φ → [z / x]φ) ↔ y([y / x]φ → [z / x]φ))
128, 11bitri 240 . . . . 5 (x(φ → [z / x]φ) ↔ y([y / x]φ → [z / x]φ))
1312albii 1566 . . . 4 (zx(φ → [z / x]φ) ↔ zy([y / x]φ → [z / x]φ))
14 alcom 1737 . . . 4 (zy([y / x]φ → [z / x]φ) ↔ yz([y / x]φ → [z / x]φ))
156, 13, 143bitri 262 . . 3 (Ⅎxφyz([y / x]φ → [z / x]φ))
16 sbhb 2107 . . . . . 6 ((φxφ) ↔ y(φ → [y / x]φ))
1716albii 1566 . . . . 5 (x(φxφ) ↔ xy(φ → [y / x]φ))
18 alcom 1737 . . . . 5 (xy(φ → [y / x]φ) ↔ yx(φ → [y / x]φ))
192, 17, 183bitri 262 . . . 4 (Ⅎxφyx(φ → [y / x]φ))
20 nfv 1619 . . . . . . 7 z(φ → [y / x]φ)
2120sb8 2092 . . . . . 6 (x(φ → [y / x]φ) ↔ z[z / x](φ → [y / x]φ))
22 nfs1v 2106 . . . . . . . 8 x[y / x]φ
2322sblim 2068 . . . . . . 7 ([z / x](φ → [y / x]φ) ↔ ([z / x]φ → [y / x]φ))
2423albii 1566 . . . . . 6 (z[z / x](φ → [y / x]φ) ↔ z([z / x]φ → [y / x]φ))
2521, 24bitri 240 . . . . 5 (x(φ → [y / x]φ) ↔ z([z / x]φ → [y / x]φ))
2625albii 1566 . . . 4 (yx(φ → [y / x]φ) ↔ yz([z / x]φ → [y / x]φ))
2719, 26bitri 240 . . 3 (Ⅎxφyz([z / x]φ → [y / x]φ))
2815, 27anbi12i 678 . 2 ((Ⅎxφ xφ) ↔ (yz([y / x]φ → [z / x]φ) yz([z / x]φ → [y / x]φ)))
29 anidm 625 . 2 ((Ⅎxφ xφ) ↔ Ⅎxφ)
301, 28, 293bitr2ri 265 1 (Ⅎxφyz([y / x]φ ↔ [z / x]φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbnfc2  3196
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