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Mirrors > Home > ILE Home > Th. List > syl3c | GIF version |
Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.) |
Ref | Expression |
---|---|
syl3c.1 | ⊢ (𝜑 → 𝜓) |
syl3c.2 | ⊢ (𝜑 → 𝜒) |
syl3c.3 | ⊢ (𝜑 → 𝜃) |
syl3c.4 | ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) |
Ref | Expression |
---|---|
syl3c | ⊢ (𝜑 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3c.3 | . 2 ⊢ (𝜑 → 𝜃) | |
2 | syl3c.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | syl3c.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
4 | syl3c.4 | . . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏))) | |
5 | 2, 3, 4 | sylc 62 | . 2 ⊢ (𝜑 → (𝜃 → 𝜏)) |
6 | 1, 5 | mpd 13 | 1 ⊢ (𝜑 → 𝜏) |
Colors of variables: wff set 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: bilukdc 1396 disjiun 3997 tfrlem1 6306 tfrcl 6362 mkvprop 7153 ccfunen 7260 caucvgprprlemval 7684 suplocsrlem 7804 peano5uzti 9357 zfz1iso 10814 lcmneg 12066 prmind2 12112 pcfac 12340 cnmpt12 13658 cnmpt22 13665 limccnp2lem 14016 2sqlem6 14327 2sqlem8 14330 sbthom 14634 |
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