Theorem axorbciffatcxorb 43498

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.)

Hypotheses

Ref	Expression
axorbciffatcxorb.1	⊢ (𝜑 ⊻ 𝜓)
axorbciffatcxorb.2	⊢ (𝜒 ↔ 𝜑)

Assertion

Ref	Expression
axorbciffatcxorb	⊢ (𝜒 ⊻ 𝜓)

Proof of Theorem axorbciffatcxorb

Step	Hyp	Ref	Expression
1		axorbciffatcxorb.1	. . . . 5 ⊢ (𝜑 ⊻ 𝜓)
2	1	axorbtnotaiffb 43496	. . . 4 ⊢ ¬ (𝜑 ↔ 𝜓)
3		xor3 387	. . . 4 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	2, 3	mpbi 233	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
5		axorbciffatcxorb.2	. . 3 ⊢ (𝜒 ↔ 𝜑)
6	4, 5	aiffnbandciffatnotciffb 43497	. 2 ⊢ ¬ (𝜒 ↔ 𝜓)
7		df-xor 1503	. 2 ⊢ ((𝜒 ⊻ 𝜓) ↔ ¬ (𝜒 ↔ 𝜓))
8	6, 7	mpbir 234	1 ⊢ (𝜒 ⊻ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1502

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 210  df-xor 1503

This theorem is referenced by:  mdandyvrx0  43574  mdandyvrx1  43575  mdandyvrx2  43576  mdandyvrx3  43577  mdandyvrx4  43578  mdandyvrx5  43579  mdandyvrx6  43580  mdandyvrx7  43581