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| Mirrors > Home > MPE Home > Th. List > sylg | Structured version Visualization version GIF version | ||
| Description: A syllogism combined with generalization. Inference associated with sylgt 1822. General form of alrimih 1824. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1824. (Revised by BJ, 4-Oct-2019.) |
| Ref | Expression |
|---|---|
| sylg.1 | ⊢ (𝜑 → ∀𝑥𝜓) |
| sylg.2 | ⊢ (𝜓 → 𝜒) |
| Ref | Expression |
|---|---|
| sylg | ⊢ (𝜑 → ∀𝑥𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylg.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜓) | |
| 2 | sylg.2 | . . 3 ⊢ (𝜓 → 𝜒) | |
| 3 | 2 | alimi 1811 | . 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → ∀𝑥𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-gen 1795 ax-4 1809 |
| This theorem is referenced by: alrimih 1824 ax9ALT 2732 raleqbidvvOLD 3335 csbied 3935 rzal 4509 ssrel 5792 ssrelOLD 5793 kmlem1 10191 bnj1476 34861 bnj1533 34866 bj-alrimd 36621 bj-exlimd 36626 bj-ax12ig 36637 axc11n11 36683 bj-modalbe 36689 bj-modal4 36715 bj-wnfanf 36720 bj-wnfenf 36721 bj-19.12 36762 bj-pm11.53vw 36777 mpobi123f 38169 mptbi12f 38173 ismnushort 44320 setrec2mpt 49216 |
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