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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 2206 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
4 | exlimd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
5 | 4 | 19.9 2195 | . 2 ⊢ (∃𝑥𝜒 ↔ 𝜒) |
6 | 3, 5 | syl6ib 252 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: exlimimdd 2209 exlimddOLD 2211 exlimdh 2289 equs5 2475 moexexlem 2704 2eu6 2737 ceqsalgALT 3528 alxfr 5298 copsex2t 5374 mosubopt 5391 ov3 7300 tz7.48-1 8068 ac6c4 9891 fsum2dlem 15113 fprod2dlem 15322 gsum2d2lem 19022 exlimim 34505 exellim 34507 wl-lem-moexsb 34685 exlimddvf 35280 fourierdlem31 42300 or2expropbi 43146 ich2exprop 43510 ichreuopeq 43512 reuopreuprim 43565 |
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