Theorem nf3an 1827

Description: If x is not free in φ, ψ, and χ, it is not free in (φ ∧ ψ ∧ χ). (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfan.1	⊢ Ⅎxφ
nfan.2	⊢ Ⅎxψ
nfan.3	⊢ Ⅎxχ

Assertion

Ref	Expression
nf3an	⊢ Ⅎx(φ ∧ ψ ∧ χ)

Proof of Theorem nf3an

Step	Hyp	Ref	Expression
1		df-3an 936	. 2 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
2		nfan.1	. . . 4 ⊢ Ⅎxφ
3		nfan.2	. . . 4 ⊢ Ⅎxψ
4	2, 3	nfan 1824	. . 3 ⊢ Ⅎx(φ ∧ ψ)
5		nfan.3	. . 3 ⊢ Ⅎxχ
6	4, 5	nfan 1824	. 2 ⊢ Ⅎx((φ ∧ ψ) ∧ χ)
7	1, 6	nfxfr 1570	1 ⊢ Ⅎx(φ ∧ ψ ∧ χ)

Syntax hints:   ∧ wa 358   ∧ w3a 934  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-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  hb3an  1830  vtocl3gaf  2923  mob  3018