Theorem bianfi 891

Description: A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)

Hypothesis

Ref	Expression
bianfi.1	⊢ ¬ φ

Assertion

Ref	Expression
bianfi	⊢ (φ ↔ (ψ ∧ φ))

Proof of Theorem bianfi

Step	Hyp	Ref	Expression
1		bianfi.1	. 2 ⊢ ¬ φ
2	1	intnan 880	. 2 ⊢ ¬ (ψ ∧ φ)
3	1, 2	2false 339	1 ⊢ (φ ↔ (ψ ∧ φ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  in0  3576