Theorem bi123imp0

Description: Similar to 3imp except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017)

	Ref	Expression
Hypothesis	bi23imp0.1	$$⊢\phi \leftrightarrow \left(\psi \leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$
Assertion	bi123imp0	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$

Proof

Step	Hyp	Ref	Expression
1		bi23imp0.1	$$⊢\phi \leftrightarrow \left(\psi \leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$
2		biimp	$$⊢\left(\psi \leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)\to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
3		biimp	$$⊢\left(\chi \leftrightarrow \theta \right)\to \left(\chi \to \theta \right)$$
4	2 3	syl6	$$⊢\left(\psi \leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)\to \left(\psi \to \left(\chi \to \theta \right)\right)$$
5	1 4	sylbi	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
6	5	3imp	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$