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Metamath Proof Explorer

Theorem abai

Description: Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993) (Proof shortened by Wolf Lammen, 7-Dec-2012)

		Ref	Expression
	Assertion	abai	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimt	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 → 𝜓 ) ) )
2	1	pm5.32i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ( 𝜑 → 𝜓 ) ) )