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	$$⊢\phi \to \psi$$
	pm5.21ni.2	$$⊢\chi \to \psi$$
	pm5.21nii.3	$$⊢\psi \to \left(\phi \leftrightarrow \chi \right)$$
Assertion	pm5.21nii	$$⊢\phi \leftrightarrow \chi$$

Proof

Step	Hyp	Ref	Expression
1		pm5.21ni.1	$$⊢\phi \to \psi$$
2		pm5.21ni.2	$$⊢\chi \to \psi$$
3		pm5.21nii.3	$$⊢\psi \to \left(\phi \leftrightarrow \chi \right)$$
4	1 2	pm5.21ni	$$⊢\neg \psi \to \left(\phi \leftrightarrow \chi \right)$$
5	3 4	pm2.61i	$$⊢\phi \leftrightarrow \chi$$