Theorem mdandyvrx2 44478

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
mdandyvrx2.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx2.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx2.3	⊢ (𝜒 ↔ 𝜑)
mdandyvrx2.4	⊢ (𝜃 ↔ 𝜓)
mdandyvrx2.5	⊢ (𝜏 ↔ 𝜑)
mdandyvrx2.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx2	⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx2

Step	Hyp	Ref	Expression
1		mdandyvrx2.1	. . . . 5 ⊢ (𝜑 ⊻ 𝜁)
2		mdandyvrx2.3	. . . . 5 ⊢ (𝜒 ↔ 𝜑)
3	1, 2	axorbciffatcxorb 44400	. . . 4 ⊢ (𝜒 ⊻ 𝜁)
4		mdandyvrx2.2	. . . . 5 ⊢ (𝜓 ⊻ 𝜎)
5		mdandyvrx2.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
6	4, 5	axorbciffatcxorb 44400	. . . 4 ⊢ (𝜃 ⊻ 𝜎)
7	3, 6	pm3.2i 471	. . 3 ⊢ ((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎))
8		mdandyvrx2.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
9	1, 8	axorbciffatcxorb 44400	. . 3 ⊢ (𝜏 ⊻ 𝜁)
10	7, 9	pm3.2i 471	. 2 ⊢ (((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁))
11		mdandyvrx2.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	1, 11	axorbciffatcxorb 44400	. 2 ⊢ (𝜂 ⊻ 𝜁)
13	10, 12	pm3.2i 471	1 ⊢ ((((𝜒 ⊻ 𝜁) ∧ (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:  mdandyvrx13  44489