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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 3048 | . 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 1332 ∈ wcel 1481 Ⅎwnfc 2269 Vcvv 2689 ⦋csb 3007 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-csb 3008 |
This theorem is referenced by: csbie 3050 csbing 3288 csbopabg 4014 pofun 4242 csbima12g 4908 csbiotag 5124 csbriotag 5750 csbov123g 5817 eqerlem 6468 zsumdc 11185 |
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