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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.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 22-Jan-2025.) |
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 | alinexa 1845 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
2 | ceqsexv.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | ceqsexv.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | notbid 317 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
5 | 2, 4 | ceqsalv 3511 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ 𝜓) |
6 | 1, 5 | bitr3i 276 | . 2 ⊢ (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ¬ 𝜓) |
7 | 6 | con4bii 320 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-clel 2810 |
This theorem is referenced by: ceqsexv2d 3528 ceqsex2v 3530 ceqsex3v 3531 gencbvex 3535 euxfr2w 3716 euxfr2 3718 inuni 5343 eqvinop 5487 elvvv 5751 dmfco 6987 fndmdif 7043 fndmin 7046 fmptco 7126 abrexco 7242 imaeqsexv 7359 imaeqexov 7644 uniuni 7748 elxp4 7912 elxp5 7913 brtpos2 8216 xpsnen 9054 xpcomco 9061 xpassen 9065 brttrcl2 9708 dfac5lem2 10118 cf0 10245 ltexprlem4 11033 pceu 16778 4sqlem12 16888 vdwapun 16906 gsumval3eu 19771 dprd2d2 19913 znleval 21109 metrest 24032 sleadd1 27469 addsuniflem 27481 addsasslem1 27483 addsasslem2 27484 mulsuniflem 27601 addsdilem1 27603 addsdilem2 27604 mulsasslem1 27615 mulsasslem2 27616 fmptcof2 31877 fpwrelmapffslem 31952 cusgredgex 34107 dfdm5 34739 dfrn5 34740 elima4 34742 brtxp 34847 brpprod 34852 elfix 34870 dfiota3 34890 brimg 34904 brapply 34905 brsuccf 34908 funpartlem 34909 brrestrict 34916 dfrecs2 34917 dfrdg4 34918 lshpsmreu 37974 isopos 38045 islpln5 38401 islvol5 38445 cdlemftr3 39431 dibelval3 40013 dicelval3 40046 mapdpglem3 40541 hdmapglem7a 40793 diophrex 41503 |
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