Theorem hbanOLD 2240

Description: Obsolete proof of hban 2128 as of 6-Oct-2021. (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	nfiOLD 1734	. . 3 ⊢ Ⅎ𝑥𝜑
3		hbOLD.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	nfiOLD 1734	. . 3 ⊢ Ⅎ𝑥𝜓
5	2, 4	nfanOLDOLD 2237	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
6	5	nfriOLD 2189	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nfOLD 1721

This theorem is referenced by: (None)