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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 2174. (Revised by GG, 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 1839 | . . 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 3518 | . . 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 206 ∧ wa 395 ∀wal 1534 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-clel 2813 |
This theorem is referenced by: ceqsexv2dOLD 3533 ceqsex2v 3535 ceqsex3v 3536 gencbvex 3540 euxfr2w 3728 euxfr2 3730 inuni 5355 eqvinop 5497 elvvv 5763 dmfco 7004 fndmdif 7061 fndmin 7064 fmptco 7148 abrexco 7263 imaeqsexvOLD 7382 imaeqexov 7670 uniuni 7780 elxp4 7944 elxp5 7945 brtpos2 8255 xpsnen 9093 xpcomco 9100 xpassen 9104 brttrcl2 9751 dfac5lem2 10161 cf0 10288 ltexprlem4 11076 pceu 16879 4sqlem12 16989 vdwapun 17007 gsumval3eu 19936 dprd2d2 20078 znleval 21590 metrest 24552 sleadd1 28036 addsuniflem 28048 addsasslem1 28050 addsasslem2 28051 mulsuniflem 28189 addsdilem1 28191 addsdilem2 28192 mulsasslem1 28203 mulsasslem2 28204 renegscl 28444 readdscl 28445 remulscl 28448 fmptcof2 32673 fpwrelmapffslem 32749 cusgredgex 35105 dfdm5 35753 dfrn5 35754 elima4 35756 brtxp 35861 brpprod 35866 elfix 35884 dfiota3 35904 brimg 35918 brapply 35919 brsuccf 35922 funpartlem 35923 brrestrict 35930 dfrecs2 35931 dfrdg4 35932 lshpsmreu 39090 isopos 39161 islpln5 39517 islvol5 39561 cdlemftr3 40547 dibelval3 41129 dicelval3 41162 mapdpglem3 41657 hdmapglem7a 41909 diophrex 42762 |
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