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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 1424 rspct 3608 zfpair 5427 gruen 10850 axpre-sup 11207 nn0lt2 12679 fzofzim 13746 ndvdssub 16443 cyccom 19234 alexsubALT 24075 clwlkclwwlklem2a 30027 erclwwlktr 30051 erclwwlkntr 30100 fmlasuc 35371 dfon2lem8 35772 prtlem15 38857 prtlem18 38859 2reuimp0 47064 |
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