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Mirrors > Home > ILE Home > Th. List > mpd3an23 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
Ref | Expression |
---|---|
mpd3an23.1 | ⊢ (𝜑 → 𝜓) |
mpd3an23.2 | ⊢ (𝜑 → 𝜒) |
mpd3an23.3 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
mpd3an23 | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | mpd3an23.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | mpd3an23.2 | . 2 ⊢ (𝜑 → 𝜒) | |
4 | mpd3an23.3 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
5 | 1, 2, 3, 4 | syl3anc 1238 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 980 |
This theorem is referenced by: exp0 10523 bcpasc 10745 bccl 10746 pw2dvds 12165 qnumdencoprm 12192 qeqnumdivden 12193 grpinvid 12929 mgpvalg 13131 mgpex 13133 opprex 13243 unitgrpid 13285 dvef 14158 |
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