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| Mirrors > Home > MPE Home > Th. List > rexlimi | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of exlimi 2217. For a version based on fewer axioms see rexlimiv 3148. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
| Ref | Expression |
|---|---|
| rexlimi.1 | ⊢ Ⅎ𝑥𝜓 |
| rexlimi.2 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
| Ref | Expression |
|---|---|
| rexlimi | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexlimi.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
| 2 | 1 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
| 3 | rexlimi.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 4 | 3 | r19.23 3256 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
| 5 | 2, 4 | mpbi 230 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 |
| 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-an 396 df-ex 1780 df-nf 1784 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: reuan 3896 triun 5274 reusv1 5397 reusv3 5405 iunopeqop 5526 tfinds 7881 fiun 7967 f1iun 7968 frpoins3xpg 8165 frpoins3xp3g 8166 iunfo 10579 iundom2g 10580 fsumcom2 15810 fprodcom2 16020 nosupbnd1 27759 nosupbnd2 27761 noinfbnd1 27774 noinfbnd2 27776 dfon2lem7 35790 finminlem 36319 r19.36vf 45141 allbutfiinf 45431 infxrunb3rnmpt 45439 hoidmvlelem1 46610 2zrngmmgm 48168 |
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