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Mirrors > Home > MPE Home > Th. List > rexlimi | Structured version Visualization version GIF version |
Description: Restricted quantifier version of exlimi 2211. For a version based on fewer axioms see rexlimiv 3149. (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 3064 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
3 | rexlimi.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | r19.23 3254 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
5 | 2, 4 | mpbi 229 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-ral 3063 df-rex 3072 |
This theorem is referenced by: reuan 3891 triun 5281 reusv1 5396 reusv3 5404 iunopeqop 5522 tfinds 7849 fiun 7929 f1iun 7930 frpoins3xpg 8126 frpoins3xp3g 8127 iunfo 10534 iundom2g 10535 fsumcom2 15720 fprodcom2 15928 nosupbnd1 27217 nosupbnd2 27219 noinfbnd1 27232 noinfbnd2 27234 dfon2lem7 34761 finminlem 35203 r19.36vf 43825 allbutfiinf 44130 infxrunb3rnmpt 44138 hoidmvlelem1 45311 2zrngmmgm 46844 |
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