Theorem biantr

Description: A transitive law of equivalence. Compare Theorem *4.22 of WhiteheadRussell p. 117. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Assertion	biantr	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\chi \leftrightarrow \psi \right)\right)\to \left(\phi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\left(\chi \leftrightarrow \psi \right)\to \left(\chi \leftrightarrow \psi \right)$$
2	1	bibi2d	$$⊢\left(\chi \leftrightarrow \psi \right)\to \left(\left(\phi \leftrightarrow \chi \right)\leftrightarrow \left(\phi \leftrightarrow \psi \right)\right)$$
3	2	biimparc	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\chi \leftrightarrow \psi \right)\right)\to \left(\phi \leftrightarrow \chi \right)$$