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	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$
	simplbiim.2	$$⊢\chi \to \theta$$
Assertion	simplbiim	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		simplbiim.1	$$⊢\phi \leftrightarrow \left(\psi \wedge \chi \right)$$
2		simplbiim.2	$$⊢\chi \to \theta$$
3	1	simprbi	$$⊢\phi \to \chi$$
4	3 2	syl	$$⊢\phi \to \theta$$