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| Mirrors > Home > MPE Home > Th. List > syldbl2 | Structured version Visualization version GIF version | ||
| Description: Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.) | 
| Ref | Expression | 
|---|---|
| syldbl2.1 | ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃)) | 
| Ref | Expression | 
|---|---|
| syldbl2 | ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | syldbl2.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃)) | |
| 2 | 1 | com12 32 | . 2 ⊢ (𝜓 → ((𝜑 ∧ 𝜓) → 𝜃)) | 
| 3 | 2 | anabsi7 671 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 | 
| 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 df-an 396 | 
| This theorem is referenced by: elpredimg 6335 rexdif1en 9199 php 9248 fpwwe2lem4 10675 elfzoextl 13761 rprmdvdspow 33562 rprmdvdsprod 33563 constrmon 33786 aks4d1p3 42080 primrootsunit1 42099 primrootlekpowne0 42107 sticksstones1 42148 sticksstones11 42158 unitscyglem2 42198 uspgrimprop 47878 | 
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