Theorem axorbciffatcxorb 43135

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 43133	. . . 4 ⊢ ¬ (𝜑 ↔ 𝜓)
3		xor3 386	. . . 4 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	2, 3	mpbi 232	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
5		axorbciffatcxorb.2	. . 3 ⊢ (𝜒 ↔ 𝜑)
6	4, 5	aiffnbandciffatnotciffb 43134	. 2 ⊢ ¬ (𝜒 ↔ 𝜓)
7		df-xor 1501	. 2 ⊢ ((𝜒 ⊻ 𝜓) ↔ ¬ (𝜒 ↔ 𝜓))
8	6, 7	mpbir 233	1 ⊢ (𝜒 ⊻ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1500

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 209  df-xor 1501

This theorem is referenced by:  mdandyvrx0  43211  mdandyvrx1  43212  mdandyvrx2  43213  mdandyvrx3  43214  mdandyvrx4  43215  mdandyvrx5  43216  mdandyvrx6  43217  mdandyvrx7  43218