Theorem hb3an 1830

Description: If x is not free in φ, ψ, and χ, it is not free in (φ ∧ ψ ∧ χ). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

Ref	Expression
hb.1	⊢ (φ → ∀xφ)
hb.2	⊢ (ψ → ∀xψ)
hb.3	⊢ (χ → ∀xχ)

Assertion

Ref	Expression
hb3an	⊢ ((φ ∧ ψ ∧ χ) → ∀x(φ ∧ ψ ∧ χ))

Proof of Theorem hb3an

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 ⊢ (φ → ∀xφ)
2	1	nfi 1551	. . 3 ⊢ Ⅎxφ
3		hb.2	. . . 4 ⊢ (ψ → ∀xψ)
4	3	nfi 1551	. . 3 ⊢ Ⅎxψ
5		hb.3	. . . 4 ⊢ (χ → ∀xχ)
6	5	nfi 1551	. . 3 ⊢ Ⅎxχ
7	2, 4, 6	nf3an 1827	. 2 ⊢ Ⅎx(φ ∧ ψ ∧ χ)
8	7	nfri 1762	1 ⊢ ((φ ∧ ψ ∧ χ) → ∀x(φ ∧ ψ ∧ χ))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934  ∀wal 1540

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by: (None)