Theorem 3anibar

Description: Remove a hypothesis from the second member of a biconditional. (Contributed by FL, 22-Jul-2008)

	Ref	Expression
Hypothesis	3anibar.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ ( 𝜒 ∧ 𝜏 ) ) )
Assertion	3anibar	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3anibar.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ ( 𝜒 ∧ 𝜏 ) ) )
2		simp3	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜒 )
3	2 1	mpbirand	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) )