Theorem biantrurd

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995) (Proof shortened by Andrew Salmon, 7-May-2011)

	Ref	Expression
Hypothesis	biantrud.1	⊢ ( 𝜑 → 𝜓 )
Assertion	biantrurd	⊢ ( 𝜑 → ( 𝜒 ↔ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biantrud.1	⊢ ( 𝜑 → 𝜓 )
2		ibar	⊢ ( 𝜓 → ( 𝜒 ↔ ( 𝜓 ∧ 𝜒 ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝜒 ↔ ( 𝜓 ∧ 𝜒 ) ) )