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Metamath Proof Explorer

Theorem pm4.71d

Description: Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of WhiteheadRussell p. 120. (Contributed by Mario Carneiro, 25-Dec-2016)

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

Proof

Step	Hyp	Ref	Expression
1		pm4.71rd.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		pm4.71	⊢ ( ( 𝜓 → 𝜒 ) ↔ ( 𝜓 ↔ ( 𝜓 ∧ 𝜒 ) ) )
3	1 2	sylib	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ∧ 𝜒 ) ) )