Theorem pm4.71i

Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of WhiteheadRussell p. 120. (Contributed by NM, 4-Jan-2004)

	Ref	Expression
Hypothesis	pm4.71i.1	⊢ ( 𝜑 → 𝜓 )
Assertion	pm4.71i	⊢ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm4.71i.1	⊢ ( 𝜑 → 𝜓 )
2		pm4.71	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )
3	1 2	mpbi	⊢ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) )