![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5852 | 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 5830 |
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 2740 df-sbc 2964 df-un 3134 df-sn 3599 df-pr 3600 df-uni 3811 df-iota 5179 df-riota 5831 |
This theorem is referenced by: eqsupti 6995 prsrriota 7787 recriota 7889 axcaucvglemval 7896 subadd 8160 divmulap 8632 flqlelt 10276 flqbi 10290 remim 10869 resqrtcl 11038 rersqrtthlem 11039 divalgmod 11932 dfgcd3 12011 bezout 12012 oddpwdclemxy 12169 qnumdenbi 12192 ismgmid 12796 isgrpinv 12926 |
Copyright terms: Public domain | W3C validator |