Theorem bianfi 854

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 838	. 2 ⊢ ¬ (𝜓 ∧ 𝜑)
3	1, 2	2false 617	1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  in0  3252  opthprc  4391