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Mirrors > Home > MPE Home > Th. List > syl3an3b | Structured version Visualization version GIF version |
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
Ref | Expression |
---|---|
syl3an3b.1 | ⊢ (𝜑 ↔ 𝜃) |
syl3an3b.2 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl3an3b | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an3b.1 | . . 3 ⊢ (𝜑 ↔ 𝜃) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝜑 → 𝜃) |
3 | syl3an3b.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | syl3an3 1162 | 1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 |
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 df-an 400 df-3an 1086 |
This theorem is referenced by: fresaunres1 6525 fvun2 6730 nnmsucr 8234 xrlttr 12521 iccdil 12868 icccntr 12870 hashgt23el 13781 absexpz 14657 posglbd 17752 f1omvdco3 18569 isdrngd 19520 unicld 21651 2ndcdisj2 22062 logrec 25349 cdj3lem3 30221 bnj563 32124 bnj1033 32351 lindsadd 35050 nn0rppwr 39490 stoweidlem14 42656 |
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