Theorem bianfd 892

Description: A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)

Hypothesis

Ref	Expression
bianfd.1	⊢ (φ → ¬ ψ)

Assertion

Ref	Expression
bianfd	⊢ (φ → (ψ ↔ (ψ ∧ χ)))

Proof of Theorem bianfd

Step	Hyp	Ref	Expression
1		bianfd.1	. 2 ⊢ (φ → ¬ ψ)
2	1	intnanrd 883	. 2 ⊢ (φ → ¬ (ψ ∧ χ))
3	1, 2	2falsed 340	1 ⊢ (φ → (ψ ↔ (ψ ∧ χ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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:  eueq2  3011  eueq3  3012