Theorem bianfd 889

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 874	. 2 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
3	1, 2	2falsed 650	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  eueq2dc  2765  eueq3dc  2766