Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nsyli | Structured version Visualization version GIF version |
Description: A negated syllogism inference. (Contributed by NM, 3-May-1994.) |
Ref | Expression |
---|---|
nsyli.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
nsyli.2 | ⊢ (𝜃 → ¬ 𝜒) |
Ref | Expression |
---|---|
nsyli | ⊢ (𝜑 → (𝜃 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsyli.2 | . 2 ⊢ (𝜃 → ¬ 𝜒) | |
2 | nsyli.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | con3d 155 | . 2 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓)) |
4 | 1, 3 | syl5 34 | 1 ⊢ (𝜑 → (𝜃 → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: necon3ad 3029 tz7.7 6217 onssneli 6300 tz7.48-2 8078 tz7.49 8081 php 8701 nndomo 8712 elirrv 9060 setind 9176 zorn2lem3 9920 alephval2 9994 inar1 10197 dfon2lem6 33033 sltres 33169 finminlem 33666 onint1 33797 poimirlem4 34911 nndomog 39917 gneispace 40504 |
Copyright terms: Public domain | W3C validator |