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Theorem nfexd 1854
 Description: If x is not free in φ, it is not free in ∃yφ. (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
nfald.1 yφ
nfald.2 (φ → Ⅎxψ)
Assertion
Ref Expression
nfexd (φ → Ⅎxyψ)

Proof of Theorem nfexd
StepHypRef Expression
1 df-ex 1542 . 2 (yψ ↔ ¬ y ¬ ψ)
2 nfald.1 . . . 4 yφ
3 nfald.2 . . . . 5 (φ → Ⅎxψ)
43nfnd 1791 . . . 4 (φ → Ⅎx ¬ ψ)
52, 4nfald 1852 . . 3 (φ → Ⅎxy ¬ ψ)
65nfnd 1791 . 2 (φ → Ⅎx ¬ y ¬ ψ)
71, 6nfxfrd 1571 1 (φ → Ⅎxyψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  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:  nfeld  2504
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