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Mirrors > Home > ILE Home > Th. List > cnveqi | GIF version |
Description: Equality inference for converse. (Contributed by NM, 23-Dec-2008.) |
Ref | Expression |
---|---|
cnveqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
cnveqi | ⊢ ◡𝐴 = ◡𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | cnveq 4777 | . 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 = ◡𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1343 ◡ccnv 4602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-in 3121 df-ss 3128 df-br 3982 df-opab 4043 df-cnv 4611 |
This theorem is referenced by: mptcnv 5005 cnvxp 5021 xp0 5022 imainrect 5048 cnvcnv 5055 mptpreima 5096 co01 5117 coi2 5119 cocnvres 5127 fcoi1 5367 fun11iun 5452 f1ocnvd 6039 cnvoprab 6198 f1od2 6199 mapsncnv 6657 sbthlemi8 6925 caseinj 7050 djuinj 7067 fisumcom2 11375 fprodcom2fi 11563 |
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