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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 2123 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
4 | exlimd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
5 | 4 | 19.9 2110 | . 2 ⊢ (∃𝑥𝜒 ↔ 𝜒) |
6 | 3, 5 | syl6ib 241 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1744 Ⅎwnf 1748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-ex 1745 df-nf 1750 |
This theorem is referenced by: exlimdd 2126 exlimdh 2187 equs5 2379 moexex 2570 2eu6 2587 exists2 2591 ceqsalgALT 3262 alxfr 4908 copsex2t 4986 mosubopt 5001 ovmpt2df 6834 ov3 6839 tz7.48-1 7583 ac6c4 9341 fsum2dlem 14545 fprod2dlem 14754 gsum2d2lem 18418 padct 29625 exlimim 33319 exellim 33322 wl-lem-moexsb 33480 exlimddvf 34056 stoweidlem27 40562 fourierdlem31 40673 intsaluni 40865 isomenndlem 41065 |
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