Theorem aiffnbandciffatnotciffb

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

	Ref	Expression
Hypotheses	aiffnbandciffatnotciffb.1	⊢ ( 𝜑 ↔ ¬ 𝜓 )
	aiffnbandciffatnotciffb.2	⊢ ( 𝜒 ↔ 𝜑 )
Assertion	aiffnbandciffatnotciffb	⊢ ¬ ( 𝜒 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		aiffnbandciffatnotciffb.1	⊢ ( 𝜑 ↔ ¬ 𝜓 )
2		aiffnbandciffatnotciffb.2	⊢ ( 𝜒 ↔ 𝜑 )
3	2 1	bitri	⊢ ( 𝜒 ↔ ¬ 𝜓 )
4		xor3	⊢ ( ¬ ( 𝜒 ↔ 𝜓 ) ↔ ( 𝜒 ↔ ¬ 𝜓 ) )
5	3 4	mpbir	⊢ ¬ ( 𝜒 ↔ 𝜓 )