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Mirrors > Home > MPE Home > Th. List > nsyl5 | Structured version Visualization version GIF version |
Description: A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.) |
Ref | Expression |
---|---|
nsyl4.1 | ⊢ (𝜑 → 𝜓) |
nsyl4.2 | ⊢ (¬ 𝜑 → 𝜒) |
Ref | Expression |
---|---|
nsyl5 | ⊢ (¬ 𝜓 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsyl4.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | nsyl4.2 | . . 3 ⊢ (¬ 𝜑 → 𝜒) | |
3 | 1, 2 | nsyl4 158 | . 2 ⊢ (¬ 𝜒 → 𝜓) |
4 | 3 | con1i 147 | 1 ⊢ (¬ 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 is referenced by: pm5.55 948 euor2 2610 moanimlem 2615 moexexlem 2623 eueq3 3708 opprc1 4898 opprc2 4899 mosubopt 5511 nfvres 6933 fvco4i 6993 fvmptex 7013 fvopab4ndm 7028 ressnop0 7151 csbriota 7381 ovprc 7447 ovprc1 7448 ovprc2 7449 ndmovass 7595 ndmovdistr 7596 extmptsuppeq 8173 funsssuppss 8175 eceqoveq 8816 supval2 9450 axpowndlem3 10594 adderpq 10951 mulerpq 10952 fzoval 13633 swrdnznd 14592 pfxnd0 14638 grpidval 18580 tgdif0 22495 resstopn 22690 rdgprc0 34765 bj-projval 35877 wl-nax6im 36387 itg2addnclem3 36541 or3or 42774 ndmafv 45848 nfunsnafv 45850 afvnufveq 45855 aovprc 45896 ndmaovass 45914 ndmaovdistr 45915 tz6.12-2-afv2 45945 naryfval 47314 naryfvalixp 47315 |
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