Theorem bija

Description: Combine antecedents into a single biconditional. This inference, reminiscent of ja , is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im and pm5.21im ). (Contributed by Wolf Lammen, 13-May-2013)

	Ref	Expression
Hypotheses	bija.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	bija.2	$$⊢\neg \phi \to \left(\neg \psi \to \chi \right)$$
Assertion	bija	$$⊢\left(\phi \leftrightarrow \psi \right)\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		bija.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		bija.2	$$⊢\neg \phi \to \left(\neg \psi \to \chi \right)$$
3		biimpr	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$
4	3 1	syli	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \chi \right)$$
5		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
6	5	con3d	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\neg \psi \to \neg \phi \right)$$
7	6 2	syli	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\neg \psi \to \chi \right)$$
8	4 7	pm2.61d	$$⊢\left(\phi \leftrightarrow \psi \right)\to \chi$$