Theorem nfnan 1825

Description: If x is not free in φ and ψ, then it is not free in (φ ⊼ ψ). (Contributed by Scott Fenton, 2-Jan-2018.)

Hypotheses

Ref	Expression
nfan.1	⊢ Ⅎxφ
nfan.2	⊢ Ⅎxψ

Assertion

Ref	Expression
nfnan	⊢ Ⅎx(φ ⊼ ψ)

Proof of Theorem nfnan

Step	Hyp	Ref	Expression
1		df-nan 1288	. 2 ⊢ ((φ ⊼ ψ) ↔ ¬ (φ ∧ ψ))
2		nfan.1	. . . 4 ⊢ Ⅎxφ
3		nfan.2	. . . 4 ⊢ Ⅎxψ
4	2, 3	nfan 1824	. . 3 ⊢ Ⅎx(φ ∧ ψ)
5	4	nfn 1793	. 2 ⊢ Ⅎx ¬ (φ ∧ ψ)
6	1, 5	nfxfr 1570	1 ⊢ Ⅎx(φ ⊼ ψ)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 358   ⊼ wnan 1287  Ⅎwnf 1544

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by:  nfnin  3229