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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 4785 | . 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 1348 ◡ccnv 4610 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 df-br 3990 df-opab 4051 df-cnv 4619 |
This theorem is referenced by: cnvsng 5096 cores2 5123 suppssof1 6078 2ndval2 6135 2nd1st 6159 cnvf1olem 6203 brtpos2 6230 dftpos4 6242 tpostpos 6243 tposf12 6248 xpcomco 6804 infeq123d 6993 fsumcnv 11400 fprodcnv 11588 ennnfonelemf1 12373 grpinvcnv 12767 txswaphmeolem 13114 |
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