Theorem hbanOLD 2337

Description: Obsolete version of hban 1906 as of 22-Oct-2026. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem hbanOLD

Step	Hyp	Ref	Expression
1		hbOLD.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2186	. . 3 ⊢ Ⅎ𝑥𝜑
3		hbOLD.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nf5i 2186	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfan 1934	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nf5ri 2235	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by: (None)