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Mirrors > Home > MPE Home > Th. List > reu4 | Structured version Visualization version GIF version |
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
rmo4.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reu4 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reu5 3136 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | |
2 | rmo4.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | rmo4 3366 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
4 | 3 | anbi2i 726 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) |
5 | 1, 4 | bitri 263 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wral 2896 ∃wrex 2897 ∃!wreu 2898 ∃*wrmo 2899 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 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-eu 2462 df-mo 2463 df-cleq 2603 df-clel 2606 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 |
This theorem is referenced by: reuind 3378 oawordeulem 7521 fin23lem23 9031 nqereu 9630 receu 10551 lbreu 10852 cju 10893 fprodser 14518 divalglem9 14962 ndvdssub 14971 qredeu 15210 pj1eu 17932 efgredeu 17988 lspsneu 18944 qtopeu 21329 qtophmeo 21430 minveclem7 23014 ig1peu 23735 coeeu 23785 plydivalg 23858 hlcgreu 25313 mirreu3 25349 trgcopyeu 25498 axcontlem2 25645 usgra2edg1 25912 usgraedgreu 25917 nbgraf1olem1 25970 4cycl2vnunb 26544 frg2wot1 26584 minvecolem7 27123 hlimreui 27480 riesz4i 28306 cdjreui 28675 xreceu 28961 cvmseu 30512 nocvxmin 31090 segconeu 31288 outsideofeu 31408 poimirlem4 32583 bfp 32793 exidu1 32825 rngoideu 32872 lshpsmreu 33414 cdleme 34866 lcfl7N 35808 mapdpg 36013 hdmap14lem6 36183 mpaaeu 36739 icceuelpart 39974 umgr2edg1 40438 umgr2edgneu 40441 usgredgreu 40445 uspgredg2vtxeu 40447 4cycl2vnunb-av 41460 frgr2wwlk1 41494 frgr2wwlkeqm 41496 |
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