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Mirrors > Home > ILE Home > Th. List > cnveqd | GIF version |
Description: Equality deduction for converse. (Contributed by NM, 6-Dec-2013.) |
Ref | Expression |
---|---|
cnveqd.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
cnveqd | ⊢ (φ → ◡A = ◡B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveqd.1 | . 2 ⊢ (φ → A = B) | |
2 | cnveq 4452 | . 2 ⊢ (A = B → ◡A = ◡B) | |
3 | 1, 2 | syl 14 | 1 ⊢ (φ → ◡A = ◡B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ◡ccnv 4287 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-in 2918 df-ss 2925 df-br 3756 df-opab 3810 df-cnv 4296 |
This theorem is referenced by: cnvsng 4749 cores2 4776 suppssof1 5670 2ndval2 5725 2nd1st 5748 cnvf1olem 5787 brtpos2 5807 dftpos4 5819 tpostpos 5820 tposf12 5825 xpcomco 6236 |
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