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Mirrors > Home > MPE Home > Th. List > syl7bi | Structured version Visualization version GIF version |
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
syl7bi.1 | ⊢ (𝜑 ↔ 𝜓) |
syl7bi.2 | ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏))) |
Ref | Expression |
---|---|
syl7bi | ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl7bi.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝜑 → 𝜓) |
3 | syl7bi.2 | . 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏))) | |
4 | 2, 3 | syl7 74 | 1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 |
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 210 |
This theorem is referenced by: 3jao 1422 rspct 3557 zfpair 5287 gruen 10223 axpre-sup 10580 nn0lt2 12033 fzofzim 13079 ndvdssub 15750 cyccom 18338 alexsubALT 22656 clwlkclwwlklem2a 27783 erclwwlktr 27807 erclwwlkntr 27856 fmlasuc 32746 dfon2lem8 33148 prtlem15 36171 prtlem18 36173 2reuimp0 43670 |
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