Theorem mdandyvrx3 44490

Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Hypotheses

Ref	Expression
mdandyvrx3.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx3.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx3.3	⊢ (𝜒 ↔ 𝜓)
mdandyvrx3.4	⊢ (𝜃 ↔ 𝜓)
mdandyvrx3.5	⊢ (𝜏 ↔ 𝜑)
mdandyvrx3.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx3	⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx3

Step	Hyp	Ref	Expression
1		mdandyvrx3.2	. . . . 5 ⊢ (𝜓 ⊻ 𝜎)
2		mdandyvrx3.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
3	1, 2	axorbciffatcxorb 44411	. . . 4 ⊢ (𝜒 ⊻ 𝜎)
4		mdandyvrx3.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
5	1, 4	axorbciffatcxorb 44411	. . . 4 ⊢ (𝜃 ⊻ 𝜎)
6	3, 5	pm3.2i 471	. . 3 ⊢ ((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎))
7		mdandyvrx3.1	. . . 4 ⊢ (𝜑 ⊻ 𝜁)
8		mdandyvrx3.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
9	7, 8	axorbciffatcxorb 44411	. . 3 ⊢ (𝜏 ⊻ 𝜁)
10	6, 9	pm3.2i 471	. 2 ⊢ (((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁))
11		mdandyvrx3.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	7, 11	axorbciffatcxorb 44411	. 2 ⊢ (𝜂 ⊻ 𝜁)
13	10, 12	pm3.2i 471	1 ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Syntax hints:   ↔ wb 205   ∧ wa 396   ⊻ wxo 1506

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  df-an 397  df-xor 1507

This theorem is referenced by:  mdandyvrx12  44499