Theorem mdandyvrx7 47005

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
mdandyvrx7.1	⊢ (𝜑 ⊻ 𝜁)
mdandyvrx7.2	⊢ (𝜓 ⊻ 𝜎)
mdandyvrx7.3	⊢ (𝜒 ↔ 𝜓)
mdandyvrx7.4	⊢ (𝜃 ↔ 𝜓)
mdandyvrx7.5	⊢ (𝜏 ↔ 𝜓)
mdandyvrx7.6	⊢ (𝜂 ↔ 𝜑)

Assertion

Ref	Expression
mdandyvrx7	⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁))

Proof of Theorem mdandyvrx7

Step	Hyp	Ref	Expression
1		mdandyvrx7.2	. . . . 5 ⊢ (𝜓 ⊻ 𝜎)
2		mdandyvrx7.3	. . . . 5 ⊢ (𝜒 ↔ 𝜓)
3	1, 2	axorbciffatcxorb 46922	. . . 4 ⊢ (𝜒 ⊻ 𝜎)
4		mdandyvrx7.4	. . . . 5 ⊢ (𝜃 ↔ 𝜓)
5	1, 4	axorbciffatcxorb 46922	. . . 4 ⊢ (𝜃 ⊻ 𝜎)
6	3, 5	pm3.2i 470	. . 3 ⊢ ((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎))
7		mdandyvrx7.5	. . . 4 ⊢ (𝜏 ↔ 𝜓)
8	1, 7	axorbciffatcxorb 46922	. . 3 ⊢ (𝜏 ⊻ 𝜎)
9	6, 8	pm3.2i 470	. 2 ⊢ (((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎))
10		mdandyvrx7.1	. . 3 ⊢ (𝜑 ⊻ 𝜁)
11		mdandyvrx7.6	. . 3 ⊢ (𝜂 ↔ 𝜑)
12	10, 11	axorbciffatcxorb 46922	. 2 ⊢ (𝜂 ⊻ 𝜁)
13	9, 12	pm3.2i 470	1 ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁))

Syntax hints:   ↔ wb 206   ∧ wa 395   ⊻ wxo 1510

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 207  df-an 396  df-xor 1511

This theorem is referenced by:  mdandyvrx8  47006