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Mirrors > Home > MPE Home > Th. List > riota2 | Structured version Visualization version GIF version |
Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
riota2.1 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riota2 | ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2913 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfv 1995 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | riota2.1 | . 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | riota2f 6776 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃!wreu 3063 ℩crio 6754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-reu 3068 df-v 3353 df-sbc 3589 df-un 3729 df-sn 4318 df-pr 4320 df-uni 4576 df-iota 5995 df-riota 6755 |
This theorem is referenced by: eqsup 8519 sup0 8529 fin23lem22 9352 subadd 10487 divmul 10891 fllelt 12807 flflp1 12817 flval2 12824 flbi 12826 remim 14066 resqrtcl 14203 resqrtthlem 14204 sqrtneg 14217 sqrtthlem 14311 divalgmod 15338 divalgmodOLD 15339 qnumdenbi 15660 catidd 16549 lubprop 17195 glbprop 17208 isglbd 17326 poslubd 17357 ismgmid 17473 isgrpinv 17681 pj1id 18320 coeeq 24204 ismir 25776 mireq 25782 ismidb 25892 islmib 25901 usgredg2vlem2 26341 frgrncvvdeqlem3 27484 frgr2wwlkeqm 27514 cnidOLD 27778 hilid 28359 pjpreeq 28598 cnvbraval 29310 cdj3lem2 29635 xdivmul 29974 cvmliftphtlem 31638 cvmlift3lem4 31643 cvmlift3lem6 31645 cvmlift3lem9 31648 scutbday 32251 scutun12 32255 scutbdaylt 32260 transportprops 32479 ltflcei 33731 cmpidelt 33991 exidresid 34011 lshpkrlem1 34920 cdlemeiota 36395 dochfl1 37287 hgmapvs 37702 wessf1ornlem 39892 fourierdlem50 40891 |
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