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Mirrors > Home > ILE Home > Th. List > csbeq2 | GIF version |
Description: Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Ref | Expression |
---|---|
csbeq2 | ⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2234 | . . . . 5 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) | |
2 | 1 | alimi 1448 | . . . 4 ⊢ (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
3 | sbcbi2 3005 | . . . 4 ⊢ (∀𝑥(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (∀𝑥 𝐵 = 𝐶 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) |
5 | 4 | abbidv 2288 | . 2 ⊢ (∀𝑥 𝐵 = 𝐶 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
6 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
7 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4g 2228 | 1 ⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 = wceq 1348 ∈ wcel 2141 {cab 2156 [wsbc 2955 ⦋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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 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-sbc 2956 df-csb 3050 |
This theorem is referenced by: prodeq2w 11519 |
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