![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cbvcsbw | GIF version |
Description: Version of cbvcsb 3074 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvcsbw.1 | ⊢ Ⅎ𝑦𝐶 |
cbvcsbw.2 | ⊢ Ⅎ𝑥𝐷 |
cbvcsbw.3 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cbvcsbw | ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvcsbw.1 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
2 | 1 | nfcri 2323 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
3 | cbvcsbw.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
4 | 3 | nfcri 2323 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
5 | cbvcsbw.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2257 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
7 | 2, 4, 6 | cbvsbcw 3002 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
8 | 7 | abbii 2303 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
9 | df-csb 3070 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
10 | df-csb 3070 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
11 | 8, 9, 10 | 3eqtr4i 2218 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 {cab 2173 Ⅎwnfc 2316 [wsbc 2974 ⦋csb 3069 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-sbc 2975 df-csb 3070 |
This theorem is referenced by: cbvprod 11579 |
Copyright terms: Public domain | W3C validator |