Theorem bianfd 537

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 492	. 2 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
3	1, 2	2falsed 379	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  eueq2  3701  eueq3  3702