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Mirrors > Home > MPE Home > Th. List > mpbidi | Structured version Visualization version GIF version |
Description: A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |
Ref | Expression |
---|---|
mpbidi.min | ⊢ (𝜃 → (𝜑 → 𝜓)) |
mpbidi.maj | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mpbidi | ⊢ (𝜃 → (𝜑 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbidi.min | . 2 ⊢ (𝜃 → (𝜑 → 𝜓)) | |
2 | mpbidi.maj | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimpd 232 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
4 | 1, 3 | sylcom 30 | 1 ⊢ (𝜃 → (𝜑 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 |
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 |
This theorem is referenced by: ralxfr2d 5276 ovmpt4g 7276 ov3 7291 omeulem2 8192 domtriomlem 9853 nsmallnq 10388 bposlem1 25868 pntrsumbnd 26150 elntg2 26779 mptsnunlem 34755 poimirlem27 35084 frege92 40656 nzss 41021 setis 45227 |
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