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Metamath Proof Explorer

Theorem simplbiim

Description: Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Wolf Lammen, 26-Mar-2022)

	Ref	Expression
Hypotheses	simplbiim.1	$⊢ φ \leftrightarrow (ψ \land χ)$
	simplbiim.2	$⊢ χ \to θ$
Assertion	simplbiim	$⊢ φ \to θ$

Proof

Step	Hyp	Ref	Expression
1		simplbiim.1	$⊢ φ \leftrightarrow (ψ \land χ)$
2		simplbiim.2	$⊢ χ \to θ$
3	1	simprbi	$⊢ φ \to χ$
4	3 2	syl	$⊢ φ \to θ$