Theorem bianfi 533

Description: A wff conjoined with falsehood is false. (Contributed by NM, 21-Jun-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 486	. 2 ⊢ ¬ (𝜓 ∧ 𝜑)
3	1, 2	2false 375	1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  in0  4395  opthprc  5749  ind1a  32844