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Mathbox for Jarvin Udandy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mdandyv3 | Structured version Visualization version GIF version |
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.) |
Ref | Expression |
---|---|
mdandyv3.1 | ⊢ (𝜑 ↔ ⊥) |
mdandyv3.2 | ⊢ (𝜓 ↔ ⊤) |
mdandyv3.3 | ⊢ (𝜒 ↔ ⊤) |
mdandyv3.4 | ⊢ (𝜃 ↔ ⊤) |
mdandyv3.5 | ⊢ (𝜏 ↔ ⊥) |
mdandyv3.6 | ⊢ (𝜂 ↔ ⊥) |
Ref | Expression |
---|---|
mdandyv3 | ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdandyv3.3 | . . . . 5 ⊢ (𝜒 ↔ ⊤) | |
2 | mdandyv3.2 | . . . . 5 ⊢ (𝜓 ↔ ⊤) | |
3 | 1, 2 | bothtbothsame 39715 | . . . 4 ⊢ (𝜒 ↔ 𝜓) |
4 | mdandyv3.4 | . . . . 5 ⊢ (𝜃 ↔ ⊤) | |
5 | 4, 2 | bothtbothsame 39715 | . . . 4 ⊢ (𝜃 ↔ 𝜓) |
6 | 3, 5 | pm3.2i 470 | . . 3 ⊢ ((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) |
7 | mdandyv3.5 | . . . 4 ⊢ (𝜏 ↔ ⊥) | |
8 | mdandyv3.1 | . . . 4 ⊢ (𝜑 ↔ ⊥) | |
9 | 7, 8 | bothfbothsame 39716 | . . 3 ⊢ (𝜏 ↔ 𝜑) |
10 | 6, 9 | pm3.2i 470 | . 2 ⊢ (((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) |
11 | mdandyv3.6 | . . 3 ⊢ (𝜂 ↔ ⊥) | |
12 | 11, 8 | bothfbothsame 39716 | . 2 ⊢ (𝜂 ↔ 𝜑) |
13 | 10, 12 | pm3.2i 470 | 1 ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ⊤wtru 1476 ⊥wfal 1480 |
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: (None) |
Copyright terms: Public domain | W3C validator |