Theorem hbanOLD 1829

Description: Obsolete proof of hban 1828 as of 2-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem hbanOLD

Step	Hyp	Ref	Expression
1		df-an 360	. 2 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
2		hb.1	. . . 4 ⊢ (φ → ∀xφ)
3		hb.2	. . . . 5 ⊢ (ψ → ∀xψ)
4	3	hbn 1776	. . . 4 ⊢ (¬ ψ → ∀x ¬ ψ)
5	2, 4	hbim 1817	. . 3 ⊢ ((φ → ¬ ψ) → ∀x(φ → ¬ ψ))
6	5	hbn 1776	. 2 ⊢ (¬ (φ → ¬ ψ) → ∀x ¬ (φ → ¬ ψ))
7	1, 6	hbxfrbi 1568	1 ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)