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Mirrors > Home > MPE Home > Th. List > ceqsexv | Structured version Visualization version GIF version |
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
Ref | Expression |
---|---|
ceqsexv.1 | ⊢ 𝐴 ∈ V |
ceqsexv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsexv | ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsexv.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | ceqsexv.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | ceqsex 3214 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: ceqsexv2d 3216 ceqsex3v 3219 gencbvex 3223 euxfr2 3358 inuni 4753 eqvinop 4881 elvvv 5101 dmfco 6182 fndmdif 6229 fndmin 6232 fmptco 6303 abrexco 6406 uniuni 6865 elxp4 7003 elxp5 7004 opabex3d 7037 opabex3 7038 fsplit 7169 brtpos2 7245 mapsnen 7920 xpsnen 7929 xpcomco 7935 xpassen 7939 dfac5lem1 8829 dfac5lem2 8830 dfac5lem3 8831 cf0 8956 ltexprlem4 9740 pceu 15389 4sqlem12 15498 vdwapun 15516 gsumval3eu 18128 dprd2d2 18266 znleval 19722 metrest 22139 fmptcof2 28839 fpwrelmapffslem 28895 dfdm5 30921 dfrn5 30922 elima4 30924 nofulllem5 31105 brtxp 31157 brpprod 31162 elfix 31180 dffix2 31182 sscoid 31190 elfuns 31192 dfiota3 31200 brimg 31214 brapply 31215 brsuccf 31218 funpartlem 31219 brrestrict 31226 dfrecs2 31227 dfrdg4 31228 lshpsmreu 33414 isopos 33485 islpln5 33839 islvol5 33883 pmapglb 34074 polval2N 34210 cdlemftr3 34871 dibelval3 35454 dicelval3 35487 mapdpglem3 35982 hdmapglem7a 36237 diophrex 36357 mapsnend 38386 opeliun2xp 41904 |
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