Theorem hb3an 2301

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	nf5i 2144	. . 3 ⊢ Ⅎ𝑥𝜑
3		hb.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2144	. . 3 ⊢ Ⅎ𝑥𝜓
5		hb.3	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
6	5	nf5i 2144	. . 3 ⊢ Ⅎ𝑥𝜒
7	2, 4, 6	nf3an 1905	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)
8	7	nf5ri 2191	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ w3a 1085  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by:  bnj982  32658  bnj1276  32694  bnj1350  32705