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Theorem nfimd 1808
 Description: If in a context x is not free in ψ and χ, it is not free in (ψ → χ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Hypotheses
Ref Expression
nfimd.1 (φ → Ⅎxψ)
nfimd.2 (φ → Ⅎxχ)
Assertion
Ref Expression
nfimd (φ → Ⅎx(ψχ))

Proof of Theorem nfimd
StepHypRef Expression
1 nfimd.1 . 2 (φ → Ⅎxψ)
2 nfimd.2 . 2 (φ → Ⅎxχ)
3 nfnf1 1790 . . . 4 xxψ
4 nfnf1 1790 . . . 4 xxχ
5 nfr 1761 . . . . . 6 (Ⅎxχ → (χxχ))
65imim2d 48 . . . . 5 (Ⅎxχ → ((ψχ) → (ψxχ)))
7 19.21t 1795 . . . . . 6 (Ⅎxψ → (x(ψχ) ↔ (ψxχ)))
87biimprd 214 . . . . 5 (Ⅎxψ → ((ψxχ) → x(ψχ)))
96, 8syl9r 67 . . . 4 (Ⅎxψ → (Ⅎxχ → ((ψχ) → x(ψχ))))
103, 4, 9alrimd 1769 . . 3 (Ⅎxψ → (Ⅎxχx((ψχ) → x(ψχ))))
11 df-nf 1545 . . 3 (Ⅎx(ψχ) ↔ x((ψχ) → x(ψχ)))
1210, 11syl6ibr 218 . 2 (Ⅎxψ → (Ⅎxχ → Ⅎx(ψχ)))
131, 2, 12sylc 56 1 (φ → Ⅎx(ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544 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-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfimOLD  1814  hbimd  1815  19.23tOLD  1819  nfand  1822  nfbid  1832  nfbidOLD  1833  nfnfOLD  1846  19.21tOLD  1863  dvelimf  1997  nfsb4t  2080  nfmod2  2217  nfrald  2665  nfifd  3685
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