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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 2854). (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 2554 | . . . . 5 ⊢ x ∈ V | |
2 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
3 | 1, 2 | op2ndd 5718 | . . . 4 ⊢ (A = 〈x, y〉 → (2nd ‘A) = y) |
4 | 3 | eqcomd 2042 | . . 3 ⊢ (A = 〈x, y〉 → y = (2nd ‘A)) |
5 | csbeq1a 2854 | . . 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 5717 | . . . 4 ⊢ (A = 〈x, y〉 → (1st ‘A) = x) |
8 | 7 | eqcomd 2042 | . . 3 ⊢ (A = 〈x, y〉 → x = (1st ‘A)) |
9 | csbeq1a 2854 | . . 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 2070 | 1 ⊢ (A = 〈x, y〉 → ⦋(1st ‘A) / x⦌⦋(2nd ‘A) / y⦌B = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ⦋csb 2846 〈cop 3370 ‘cfv 4845 1st c1st 5707 2nd c2nd 5708 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-1st 5709 df-2nd 5710 |
This theorem is referenced by: dfmpt2 5786 |
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