Theorem pm5.21nii

Description: Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999)

	Ref	Expression
Hypotheses	pm5.21ni.1	⊢ ( 𝜑 → 𝜓 )
	pm5.21ni.2	⊢ ( 𝜒 → 𝜓 )
	pm5.21nii.3	⊢ ( 𝜓 → ( 𝜑 ↔ 𝜒 ) )
Assertion	pm5.21nii	⊢ ( 𝜑 ↔ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		pm5.21ni.1	⊢ ( 𝜑 → 𝜓 )
2		pm5.21ni.2	⊢ ( 𝜒 → 𝜓 )
3		pm5.21nii.3	⊢ ( 𝜓 → ( 𝜑 ↔ 𝜒 ) )
4	1 2	pm5.21ni	⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) )
5	3 4	pm2.61i	⊢ ( 𝜑 ↔ 𝜒 )