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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 2779 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 2324 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
3 | nfcsb1v 3082 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
4 | csbhypf.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
5 | 3, 4 | nfeq 2320 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶 |
6 | 2, 5 | nfim 1565 | . 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
7 | eqeq1 2177 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | csbeq1a 3058 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 8 | eqeq1d 2179 | . . 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 1750 | 1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 Ⅎwnfc 2299 ⦋csb 3049 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: disji2 3980 tfisi 4569 |
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