Theorem mdandyvrx4 44367

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
mdandyvrx4.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx4.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx4.3	⊢ (𝜒 ↔ 𝜑)
mdandyvrx4.4	⊢ (𝜃 ↔ 𝜑)
mdandyvrx4.5	⊢ (𝜏 ↔ 𝜓)
mdandyvrx4.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx4	⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx4

Step	Hyp	Ref	Expression
1		mdandyvrx4.1	. . . . 5 ⊢ (𝜑 ⊻ 𝜁)
2		mdandyvrx4.3	. . . . 5 ⊢ (𝜒 ↔ 𝜑)
3	1, 2	axorbciffatcxorb 44287	. . . 4 ⊢ (𝜒 ⊻ 𝜁)
4		mdandyvrx4.4	. . . . 5 ⊢ (𝜃 ↔ 𝜑)
5	1, 4	axorbciffatcxorb 44287	. . . 4 ⊢ (𝜃 ⊻ 𝜁)
6	3, 5	pm3.2i 470	. . 3 ⊢ ((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁))
7		mdandyvrx4.2	. . . 4 ⊢ (𝜓 ⊻ 𝜎)
8		mdandyvrx4.5	. . . 4 ⊢ (𝜏 ↔ 𝜓)
9	7, 8	axorbciffatcxorb 44287	. . 3 ⊢ (𝜏 ⊻ 𝜎)
10	6, 9	pm3.2i 470	. 2 ⊢ (((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎))
11		mdandyvrx4.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	1, 11	axorbciffatcxorb 44287	. 2 ⊢ (𝜂 ⊻ 𝜁)
13	10, 12	pm3.2i 470	1 ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁))

Syntax hints:   ↔ wb 205   ∧ wa 395   ⊻ wxo 1503

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 396  df-xor 1504

This theorem is referenced by:  mdandyvrx11  44374