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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 4783 | . 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 4608 |
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 3988 df-opab 4049 df-cnv 4617 |
This theorem is referenced by: cnvsng 5094 cores2 5121 suppssof1 6076 2ndval2 6133 2nd1st 6157 cnvf1olem 6201 brtpos2 6228 dftpos4 6240 tpostpos 6241 tposf12 6246 xpcomco 6802 infeq123d 6991 fsumcnv 11393 fprodcnv 11581 ennnfonelemf1 12366 txswaphmeolem 13079 |
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