Theorem pm5.16

Description: Theorem *5.16 of WhiteheadRussell p. 124. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 17-Oct-2013)

	Ref	Expression
Assertion	pm5.16	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\phi \leftrightarrow \neg \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.18	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \neg \left(\phi \leftrightarrow \neg \psi \right)$$
2	1	biimpi	$$⊢\left(\phi \leftrightarrow \psi \right)\to \neg \left(\phi \leftrightarrow \neg \psi \right)$$
3		imnan	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\to \neg \left(\phi \leftrightarrow \neg \psi \right)\right)\leftrightarrow \neg \left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\phi \leftrightarrow \neg \psi \right)\right)$$
4	2 3	mpbi	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\wedge \left(\phi \leftrightarrow \neg \psi \right)\right)$$