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Mirrors > Home > ILE Home > Th. List > csbhypf | GIF version |
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2735 for class substitution version. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
csbhypf.1 | ⊢ Ⅎ𝑥𝐴 |
csbhypf.2 | ⊢ Ⅎ𝑥𝐶 |
csbhypf.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbhypf | ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbhypf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfeq2 2293 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
3 | nfcsb1v 3035 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
4 | csbhypf.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
5 | 3, 4 | nfeq 2289 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶 |
6 | 2, 5 | nfim 1551 | . 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
7 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | csbeq1a 3012 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 8 | eqeq1d 2148 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)) |
10 | 7, 9 | imbi12d 233 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶))) |
11 | csbhypf.3 | . 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
12 | 6, 10, 11 | chvar 1730 | 1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnfc 2268 ⦋csb 3003 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-sbc 2910 df-csb 3004 |
This theorem is referenced by: disji2 3922 tfisi 4501 |
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