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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 2172. (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 1846 | . . 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 318 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
5 | 2, 4 | ceqsalv 3512 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ 𝜓) |
6 | 1, 5 | bitr3i 277 | . 2 ⊢ (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ¬ 𝜓) |
7 | 6 | con4bii 321 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 |
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-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-clel 2811 |
This theorem is referenced by: ceqsexv2d 3529 ceqsex2v 3531 ceqsex3v 3532 gencbvex 3536 euxfr2w 3717 euxfr2 3719 inuni 5344 eqvinop 5488 elvvv 5752 dmfco 6988 fndmdif 7044 fndmin 7047 fmptco 7127 abrexco 7243 imaeqsexv 7360 imaeqexov 7645 uniuni 7749 elxp4 7913 elxp5 7914 brtpos2 8217 xpsnen 9055 xpcomco 9062 xpassen 9066 brttrcl2 9709 dfac5lem2 10119 cf0 10246 ltexprlem4 11034 pceu 16779 4sqlem12 16889 vdwapun 16907 gsumval3eu 19772 dprd2d2 19914 znleval 21110 metrest 24033 sleadd1 27472 addsuniflem 27484 addsasslem1 27486 addsasslem2 27487 mulsuniflem 27604 addsdilem1 27606 addsdilem2 27607 mulsasslem1 27618 mulsasslem2 27619 fmptcof2 31882 fpwrelmapffslem 31957 cusgredgex 34112 dfdm5 34744 dfrn5 34745 elima4 34747 brtxp 34852 brpprod 34857 elfix 34875 dfiota3 34895 brimg 34909 brapply 34910 brsuccf 34913 funpartlem 34914 brrestrict 34921 dfrecs2 34922 dfrdg4 34923 lshpsmreu 37979 isopos 38050 islpln5 38406 islvol5 38450 cdlemftr3 39436 dibelval3 40018 dicelval3 40051 mapdpglem3 40546 hdmapglem7a 40798 diophrex 41513 |
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