Mathbox for Jarvin Udandy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mdandyvrx3 | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
mdandyvrx3 | ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdandyvrx3.2 | . . . . 5 ⊢ (𝜓 ⊻ 𝜎) | |
2 | mdandyvrx3.3 | . . . . 5 ⊢ (𝜒 ↔ 𝜓) | |
3 | 1, 2 | axorbciffatcxorb 44368 | . . . 4 ⊢ (𝜒 ⊻ 𝜎) |
4 | mdandyvrx3.4 | . . . . 5 ⊢ (𝜃 ↔ 𝜓) | |
5 | 1, 4 | axorbciffatcxorb 44368 | . . . 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 44368 | . . 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 44368 | . 2 ⊢ (𝜂 ⊻ 𝜁) |
13 | 10, 12 | pm3.2i 471 | 1 ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) |
Colors of variables: wff setvar class |
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 44456 |
Copyright terms: Public domain | W3C validator |