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Mirrors > Home > MPE Home > Th. List > jcn | Structured version Visualization version GIF version |
Description: Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
jcn.1 | ⊢ (𝜑 → 𝜓) |
jcn.2 | ⊢ (𝜑 → ¬ 𝜒) |
Ref | Expression |
---|---|
jcn | ⊢ (𝜑 → ¬ (𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jcn.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | jcn.2 | . . 3 ⊢ (𝜑 → ¬ 𝜒) | |
3 | 1, 2 | jc 164 | . 2 ⊢ (𝜑 → ¬ (𝜓 → ¬ ¬ 𝜒)) |
4 | notnotb 316 | . . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒) | |
5 | 4 | imbi2i 337 | . 2 ⊢ ((𝜓 → 𝜒) ↔ (𝜓 → ¬ ¬ 𝜒)) |
6 | 3, 5 | sylnibr 330 | 1 ⊢ (𝜑 → ¬ (𝜓 → 𝜒)) |
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 depends on definitions: df-bi 208 |
This theorem is referenced by: norassOLD 1525 isf34lem4 9787 strlem6 29960 hstrlem6 29968 nn0prpw 33568 unblimceq0 33743 relexpmulg 39933 limcrecl 41786 ichnreuop 43511 |
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