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Metamath Proof Explorer

Theorem pm4.71

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of WhiteheadRussell p. 120. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 2-Dec-2012)

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

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2	1	biantru	⊢ ( ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ) )
3		anclb	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
4		dfbi2	⊢ ( ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ) )
5	2 3 4	3bitr4i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )