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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 3115 | . 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 1364 ∈ wcel 2160 Ⅎwnfc 2319 Vcvv 2752 ⦋csb 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sbc 2978 df-csb 3073 |
This theorem is referenced by: csbie 3117 csbing 3357 csbopabg 4096 pofun 4330 csbima12g 5007 csbiotag 5228 csbriotag 5864 csbov123g 5934 eqerlem 6590 zsumdc 11424 |
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