Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > csbopeq1a | GIF version |
Description: Equality theorem for substitution of a class 𝐴 for an ordered pair 〈𝑥, 𝑦〉 in 𝐵 (analog of csbeq1a 2983). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
csbopeq1a | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 2663 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | op2ndd 6015 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) |
4 | 3 | eqcomd 2123 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) |
5 | csbeq1a 2983 | . . 3 ⊢ (𝑦 = (2nd ‘𝐴) → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵) |
7 | 1, 2 | op1std 6014 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) |
8 | 7 | eqcomd 2123 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) |
9 | csbeq1a 2983 | . . 3 ⊢ (𝑥 = (1st ‘𝐴) → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵) | |
10 | 8, 9 | syl 14 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵) |
11 | 6, 10 | eqtr2d 2151 | 1 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ⦋csb 2975 〈cop 3500 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 |
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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: dfmpo 6088 f1od2 6100 |
Copyright terms: Public domain | W3C validator |