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Theorem nfald 1852
 Description: If x is not free in φ, it is not free in ∀yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1 yφ
nfald.2 (φ → Ⅎxψ)
Assertion
Ref Expression
nfald (φ → Ⅎxyψ)

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . 3 yφ
2 nfald.2 . . 3 (φ → Ⅎxψ)
31, 2alrimi 1765 . 2 (φyxψ)
4 nfnf1 1790 . . . 4 xxψ
54nfal 1842 . . 3 xyxψ
6 hba1 1786 . . . 4 (yxψyyxψ)
7 sp 1747 . . . . 5 (yxψ → Ⅎxψ)
87nfrd 1763 . . . 4 (yxψ → (ψxψ))
96, 8hbald 1740 . . 3 (yxψ → (yψxyψ))
105, 9nfd 1766 . 2 (yxψ → Ⅎxyψ)
113, 10syl 15 1 (φ → Ⅎxyψ)
 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-7 1734  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfexd  1854  nfald2  1972  nfsb4t  2080  nfeqd  2503
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