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Mirrors > Home > ILE Home > Th. List > csbief | GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbief.1 | ⊢ 𝐴 ∈ V |
csbief.2 | ⊢ Ⅎ𝑥𝐶 |
csbief.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbief | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbief.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | csbief.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) |
4 | csbief.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 3013 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 Ⅎwnfc 2245 Vcvv 2660 ⦋csb 2975 |
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-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sbc 2883 df-csb 2976 |
This theorem is referenced by: csbie 3015 csbing 3253 csbopabg 3976 pofun 4204 csbima12g 4870 csbiotag 5086 csbriotag 5710 csbov123g 5777 eqerlem 6428 zsumdc 11121 |
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