Mathbox for Jarvin Udandy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mdandyvr2 | Structured version Visualization version GIF version |
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.) |
Ref | Expression |
---|---|
mdandyvr2.1 | ⊢ (𝜑 ↔ 𝜁) |
mdandyvr2.2 | ⊢ (𝜓 ↔ 𝜎) |
mdandyvr2.3 | ⊢ (𝜒 ↔ 𝜑) |
mdandyvr2.4 | ⊢ (𝜃 ↔ 𝜓) |
mdandyvr2.5 | ⊢ (𝜏 ↔ 𝜑) |
mdandyvr2.6 | ⊢ (𝜂 ↔ 𝜑) |
Ref | Expression |
---|---|
mdandyvr2 | ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdandyvr2.3 | . . . . 5 ⊢ (𝜒 ↔ 𝜑) | |
2 | mdandyvr2.1 | . . . . 5 ⊢ (𝜑 ↔ 𝜁) | |
3 | 1, 2 | bitri 263 | . . . 4 ⊢ (𝜒 ↔ 𝜁) |
4 | mdandyvr2.4 | . . . . 5 ⊢ (𝜃 ↔ 𝜓) | |
5 | mdandyvr2.2 | . . . . 5 ⊢ (𝜓 ↔ 𝜎) | |
6 | 4, 5 | bitri 263 | . . . 4 ⊢ (𝜃 ↔ 𝜎) |
7 | 3, 6 | pm3.2i 470 | . . 3 ⊢ ((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) |
8 | mdandyvr2.5 | . . . 4 ⊢ (𝜏 ↔ 𝜑) | |
9 | 8, 2 | bitri 263 | . . 3 ⊢ (𝜏 ↔ 𝜁) |
10 | 7, 9 | pm3.2i 470 | . 2 ⊢ (((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) |
11 | mdandyvr2.6 | . . 3 ⊢ (𝜂 ↔ 𝜑) | |
12 | 11, 2 | bitri 263 | . 2 ⊢ (𝜂 ↔ 𝜁) |
13 | 10, 12 | pm3.2i 470 | 1 ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 |
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 |
This theorem is referenced by: mdandyvr13 39794 |
Copyright terms: Public domain | W3C validator |