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Mirrors > Home > MPE Home > Th. List > imim12i | Structured version Visualization version GIF version |
Description: Inference joining two implications. Inference associated with imim12 105. Its associated inference is 3syl 18. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.) |
Ref | Expression |
---|---|
imim12i.1 | ⊢ (𝜑 → 𝜓) |
imim12i.2 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
imim12i | ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imim12i.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | imim12i.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
3 | 2 | imim2i 16 | . 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → 𝜃)) |
4 | 1, 3 | syl5 34 | 1 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: imim1i 63 dedlem0b 1042 meredith 1644 sbequ2 2241 sbequ2OLD 2242 pssnn 8951 pssnnOLD 9040 kmlem1 9906 brdom5 10285 brdom4 10286 axpowndlem2 10354 naim1 34578 naim2 34579 meran1 34600 bj-gl4 34777 bj-wnf1 34899 rp-fakeanorass 41120 fiinfi 41180 axc11next 42024 |
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