Theorem axorbciffatcxorb 47700

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 47698	. . . 4 ⊢ ¬ (𝜑 ↔ 𝜓)
3		xor3 385	. . . 4 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4	2, 3	mpbi 233	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
5		axorbciffatcxorb.2	. . 3 ⊢ (𝜒 ↔ 𝜑)
6	4, 5	aiffnbandciffatnotciffb 47699	. 2 ⊢ ¬ (𝜒 ↔ 𝜓)
7		df-xor 1542	. 2 ⊢ ((𝜒 ⊻ 𝜓) ↔ ¬ (𝜒 ↔ 𝜓))
8	6, 7	mpbir 234	1 ⊢ (𝜒 ⊻ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-xor 1542

This theorem is used by:  mdandyvrx0  47776  mdandyvrx1  47777  mdandyvrx2  47778  mdandyvrx3  47779  mdandyvrx4  47780  mdandyvrx5  47781  mdandyvrx6  47782  mdandyvrx7  47783