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	⊢ ( 𝜑 ⊻ 𝜓 )
	axorbciffatcxorb.2	⊢ ( 𝜒 ↔ 𝜑 )
Assertion	axorbciffatcxorb	⊢ ( 𝜒 ⊻ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		axorbciffatcxorb.1	⊢ ( 𝜑 ⊻ 𝜓 )
2		axorbciffatcxorb.2	⊢ ( 𝜒 ↔ 𝜑 )
3	1	axorbtnotaiffb	⊢ ¬ ( 𝜑 ↔ 𝜓 )
4		xor3	⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜑 ↔ ¬ 𝜓 ) )
5	3 4	mpbi	⊢ ( 𝜑 ↔ ¬ 𝜓 )
6	5 2	aiffnbandciffatnotciffb	⊢ ¬ ( 𝜒 ↔ 𝜓 )
7		df-xor	⊢ ( ( 𝜒 ⊻ 𝜓 ) ↔ ¬ ( 𝜒 ↔ 𝜓 ) )
8	6 7	mpbir	⊢ ( 𝜒 ⊻ 𝜓 )