![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sylan2d | Structured version Visualization version GIF version |
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.) |
Ref | Expression |
---|---|
sylan2d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
sylan2d.2 | ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏)) |
Ref | Expression |
---|---|
sylan2d | ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan2d.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | sylan2d.2 | . . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏)) | |
3 | 2 | ancomsd 465 | . . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) |
4 | 1, 3 | syland 603 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
5 | 4 | ancomsd 465 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 |
This theorem is referenced by: sylan2i 606 syl2and 608 swopo 5608 fprlem1 8324 wfrlem5OLD 8352 unblem1 9326 frrlem15 9795 prodgt02 12113 lo1mul 15661 infpnlem1 16944 ghmcnp 24139 ulmcaulem 26452 ulmcau 26453 shintcli 31358 ballotlemfc0 34474 ballotlemfcc 34475 btwnxfr 36038 endofsegid 36067 bj-bary1lem1 37294 matunitlindflem1 37603 ltcvrntr 39407 poml4N 39936 |
Copyright terms: Public domain | W3C validator |