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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 2216 | . 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
| 4 | exlimd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
| 5 | 4 | 19.9 2205 | . 2 ⊢ (∃𝑥𝜒 ↔ 𝜒) |
| 6 | 3, 5 | imbitrdi 251 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∃wex 1779 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: exlimimdd 2219 exlimdh 2290 equs5 2465 moexexlem 2626 2eu6 2657 ceqsalgALT 3518 alxfr 5407 copsex2t 5497 mosubopt 5515 ov3 7596 tz7.48-1 8483 ac6c4 10521 fsum2dlem 15806 fprod2dlem 16016 gsum2d2lem 19991 exlimim 37343 exellim 37345 wl-lem-moexsb 37569 exlimddvf 38128 mnringmulrcld 44247 fourierdlem31 46153 or2expropbi 47046 ich2exprop 47458 ichreuopeq 47460 reuopreuprim 47513 |
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