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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 216 | . 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 206 | 
| 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 207 | 
| This theorem is referenced by: 3jao 1426 rspct 3607 zfpair 5420 gruen 10853 axpre-sup 11210 nn0lt2 12683 fzofzim 13750 ndvdssub 16447 cyccom 19222 alexsubALT 24060 clwlkclwwlklem2a 30018 erclwwlktr 30042 erclwwlkntr 30091 fmlasuc 35392 dfon2lem8 35792 prtlem15 38877 prtlem18 38879 2reuimp0 47131 | 
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