Theorem mdandyvrx0 39797

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
mdandyvrx0.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx0.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx0.3	⊢ (𝜒 ↔ 𝜑)
mdandyvrx0.4	⊢ (𝜃 ↔ 𝜑)
mdandyvrx0.5	⊢ (𝜏 ↔ 𝜑)
mdandyvrx0.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx0	⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx0

Step	Hyp	Ref	Expression
1		mdandyvrx0.1	. . . . 5 ⊢ (𝜑 ⊻ 𝜁)
2		mdandyvrx0.3	. . . . 5 ⊢ (𝜒 ↔ 𝜑)
3	1, 2	axorbciffatcxorb 39721	. . . 4 ⊢ (𝜒 ⊻ 𝜁)
4		mdandyvrx0.4	. . . . 5 ⊢ (𝜃 ↔ 𝜑)
5	1, 4	axorbciffatcxorb 39721	. . . 4 ⊢ (𝜃 ⊻ 𝜁)
6	3, 5	pm3.2i 470	. . 3 ⊢ ((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁))
7		mdandyvrx0.5	. . . 4 ⊢ (𝜏 ↔ 𝜑)
8	1, 7	axorbciffatcxorb 39721	. . 3 ⊢ (𝜏 ⊻ 𝜁)
9	6, 8	pm3.2i 470	. 2 ⊢ (((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁))
10		mdandyvrx0.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
11	1, 10	axorbciffatcxorb 39721	. 2 ⊢ (𝜂 ⊻ 𝜁)
12	9, 11	pm3.2i 470	1 ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁))

Syntax hints:   ↔ wb 195   ∧ wa 383   ⊻ wxo 1456

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 196  df-an 385  df-xor 1457

This theorem is referenced by:  mdandyvrx15  39812