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Mirrors > Home > ILE Home > Th. List > riota2 | 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 2319 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfv 1528 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | riota2.1 | . 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | riota2f 5854 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃!wreu 2457 ℩crio 5832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-reu 2462 df-v 2741 df-sbc 2965 df-un 3135 df-sn 3600 df-pr 3601 df-uni 3812 df-iota 5180 df-riota 5833 |
This theorem is referenced by: eqsupti 6997 prsrriota 7789 recriota 7891 axcaucvglemval 7898 subadd 8162 divmulap 8634 flqlelt 10278 flqbi 10292 remim 10871 resqrtcl 11040 rersqrtthlem 11041 divalgmod 11934 dfgcd3 12013 bezout 12014 oddpwdclemxy 12171 qnumdenbi 12194 ismgmid 12801 isgrpinv 12931 |
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