Theorem hban 2304

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

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

Assertion

Ref	Expression
hban	⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Proof of Theorem hban

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2146	. . 3 ⊢ Ⅎ𝑥𝜑
3		hb.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2146	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfan 1898	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nf5ri 2196	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782

This theorem is referenced by:  bnj982  34754  bnj1351  34802  bnj1352  34803  bnj1441  34816  bnj1441g  34817  copsex2b  37106  dvelimf-o  38885  ax12indalem  38901  ax12inda2ALT  38902  hbimpg  44525  hbimpgVD  44875