Theorem axorbciffatcxorb

Description: Given a is equivalent to (not b), c is equivalent to a. there exists a proof for ( c xor b ). (Contributed by Jarvin Udandy, 7-Sep-2016)

	Ref	Expression
Hypotheses	axorbciffatcxorb.1	$$⊢\left(\phi ⊻\psi \right)$$
	axorbciffatcxorb.2	$$⊢\chi \leftrightarrow \phi$$
Assertion	axorbciffatcxorb	$$⊢\left(\chi ⊻\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		axorbciffatcxorb.1	$$⊢\left(\phi ⊻\psi \right)$$
2		axorbciffatcxorb.2	$$⊢\chi \leftrightarrow \phi$$
3	1	axorbtnotaiffb	$$⊢\neg \left(\phi \leftrightarrow \psi \right)$$
4		xor3	$$⊢\neg \left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\phi \leftrightarrow \neg \psi \right)$$
5	3 4	mpbi	$$⊢\phi \leftrightarrow \neg \psi$$
6	5 2	aiffnbandciffatnotciffb	$$⊢\neg \left(\chi \leftrightarrow \psi \right)$$
7		df-xor	$$⊢\left(\chi ⊻\psi \right)\leftrightarrow \neg \left(\chi \leftrightarrow \psi \right)$$
8	6 7	mpbir	$$⊢\left(\chi ⊻\psi \right)$$