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Mirrors > Home > MPE Home > Th. List > exlimd | Structured version Visualization version GIF version |
Description: Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
Ref | Expression |
---|---|
exlimd.1 | ⊢ Ⅎ𝑥𝜑 |
exlimd.2 | ⊢ Ⅎ𝑥𝜒 |
exlimd.3 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
exlimd | ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | exlimd.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | eximd 2240 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
4 | exlimd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
5 | 4 | 19.9 2227 | . 2 ⊢ (∃𝑥𝜒 ↔ 𝜒) |
6 | 3, 5 | syl6ib 241 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1851 Ⅎwnf 1855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-12 2202 |
This theorem depends on definitions: df-bi 197 df-ex 1852 df-nf 1857 |
This theorem is referenced by: exlimdd 2243 exlimdh 2313 equs5 2496 moexex 2689 2eu6 2706 exists2 2710 ceqsalgALT 3380 alxfr 5006 copsex2t 5084 mosubopt 5103 ovmpt2df 6938 ov3 6943 tz7.48-1 7690 ac6c4 9504 fsum2dlem 14708 fprod2dlem 14916 gsum2d2lem 18578 padct 29831 exlimim 33519 exellim 33522 wl-lem-moexsb 33677 exlimddvf 34251 stoweidlem27 40755 fourierdlem31 40866 intsaluni 41058 isomenndlem 41258 |
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