Theorem hban 2113

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 2011	. . 3 ⊢ Ⅎ𝑥𝜑
3		hb.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2011	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfan 1816	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nf5ri 2053	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  bnj982  30103  bnj1351  30151  bnj1352  30152  bnj1441  30165  dvelimf-o  33232  ax12indalem  33248  ax12inda2ALT  33249  hbimpg  37791  hbimpgVD  38162