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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 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 2904 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfv 1918 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | riota2.1 | . 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | riota2f 7390 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃!wreu 3375 ℩crio 7364 |
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 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-reu 3378 df-v 3477 df-un 3954 df-in 3956 df-ss 3966 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 df-riota 7365 |
This theorem is referenced by: eqsup 9451 sup0 9461 ttrcltr 9711 fin23lem22 10322 subadd 11463 divmul 11875 fllelt 13762 flflp1 13772 flval2 13779 flbi 13781 remim 15064 resqrtcl 15200 resqrtthlem 15201 sqrtneg 15214 sqrtthlem 15309 divalgmod 16349 qnumdenbi 16680 catidd 17624 lubprop 18311 glbprop 18324 poslubd 18366 isglbd 18462 ismgmid 18584 isgrpinv 18878 pj1id 19567 coeeq 25741 scutbday 27305 eqscut 27306 scutun12 27311 scutbdaylt 27319 divsmulw 27640 ismir 27910 mireq 27916 ismidb 28029 islmib 28038 usgredg2vlem2 28483 frgrncvvdeqlem3 29554 frgr2wwlkeqm 29584 cnidOLD 29835 hilid 30414 pjpreeq 30651 cnvbraval 31363 cdj3lem2 31688 xdivmul 32091 cvmliftphtlem 34308 cvmlift3lem4 34313 cvmlift3lem6 34315 cvmlift3lem9 34318 transportprops 35006 ltflcei 36476 cmpidelt 36727 exidresid 36747 lshpkrlem1 37980 cdlemeiota 39456 dochfl1 40347 hgmapvs 40762 evlsval3 41131 fsuppind 41162 renegadd 41245 resubadd 41252 addinvcom 41304 wessf1ornlem 43882 fourierdlem50 44872 |
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