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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 161 | . 2 ⊢ (𝜑 → ¬ (𝜓 → ¬ ¬ 𝜒)) |
4 | notnotb 307 | . . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒) | |
5 | 4 | imbi2i 328 | . 2 ⊢ ((𝜓 → 𝜒) ↔ (𝜓 → ¬ ¬ 𝜒)) |
6 | 3, 5 | sylnibr 321 | 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 199 |
This theorem is referenced by: isf34lem4 9597 strlem6 29814 hstrlem6 29822 nn0prpw 33198 unblimceq0 33372 relexpmulg 39424 limcrecl 41347 ichnreuop 43008 |
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