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Theorem nfnan 1943
Description: If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by Scott Fenton, 2-Jan-2018.)
Hypotheses
Ref Expression
nfan.1 𝑥𝜑
nfan.2 𝑥𝜓
Assertion
Ref Expression
nfnan 𝑥(𝜑𝜓)

Proof of Theorem nfnan
StepHypRef Expression
1 df-nan 1561 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 nfan.1 . . . 4 𝑥𝜑
3 nfan.2 . . . 4 𝑥𝜓
42, 3nfan 1941 . . 3 𝑥(𝜑𝜓)
54nfn 1897 . 2 𝑥 ¬ (𝜑𝜓)
61, 5nfxfr 1892 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wnan 1560  wnf 1821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-nan 1561  df-tru 1599  df-ex 1818  df-nf 1823
This theorem is referenced by: (None)
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