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Metamath Proof Explorer

Theorem pm4.71r

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of WhiteheadRussell p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999)

		Ref	Expression
	Assertion	pm4.71r	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm4.71	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2		ancom	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) )
3	2	bibi2i	⊢ ( ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜑 ) ) )
4	1 3	bitri	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜑 ) ) )