Theorem aiffnbandciffatnotciffb 44286

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.)

Hypotheses

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

Assertion

Ref	Expression
aiffnbandciffatnotciffb	⊢ ¬ (𝜒 ↔ 𝜓)

Proof of Theorem aiffnbandciffatnotciffb

Step	Hyp	Ref	Expression
1		aiffnbandciffatnotciffb.2	. . 3 ⊢ (𝜒 ↔ 𝜑)
2		aiffnbandciffatnotciffb.1	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
3	1, 2	bitri 274	. 2 ⊢ (𝜒 ↔ ¬ 𝜓)
4		xor3 383	. 2 ⊢ (¬ (𝜒 ↔ 𝜓) ↔ (𝜒 ↔ ¬ 𝜓))
5	3, 4	mpbir 230	1 ⊢ ¬ (𝜒 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  axorbciffatcxorb  44287