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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 ↔ ( 𝜓 ∧ 𝜒 ) )
	simplbiim.2	⊢ ( 𝜒 → 𝜃 )
Assertion	simplbiim	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		simplbiim.1	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )
2		simplbiim.2	⊢ ( 𝜒 → 𝜃 )
3	1	simprbi	⊢ ( 𝜑 → 𝜒 )
4	3 2	syl	⊢ ( 𝜑 → 𝜃 )