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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 4683 | . 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 1316 ◡ccnv 4508 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-in 3047 df-ss 3054 df-br 3900 df-opab 3960 df-cnv 4517 |
This theorem is referenced by: cnvsng 4994 cores2 5021 suppssof1 5967 2ndval2 6022 2nd1st 6046 cnvf1olem 6089 brtpos2 6116 dftpos4 6128 tpostpos 6129 tposf12 6134 xpcomco 6688 infeq123d 6871 fsumcnv 11174 ennnfonelemf1 11858 txswaphmeolem 12416 |
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