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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 4822 | . 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 = ◡𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ◡ccnv 4646 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-in 3150 df-ss 3157 df-br 4022 df-opab 4083 df-cnv 4655 |
This theorem is referenced by: mptcnv 5052 cnvxp 5068 xp0 5069 imainrect 5095 cnvcnv 5102 mptpreima 5143 co01 5164 coi2 5166 cocnvres 5174 fcoi1 5418 fun11iun 5504 f1ocnvd 6100 cnvoprab 6263 f1od2 6264 mapsncnv 6725 sbthlemi8 6997 caseinj 7122 djuinj 7139 fisumcom2 11487 fprodcom2fi 11675 |
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