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Mirrors > Home > ILE Home > Th. List > csbeq2d | GIF version |
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
csbeq2d.1 | ⊢ Ⅎ𝑥𝜑 |
csbeq2d.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbeq2d | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq2d.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | csbeq2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2107 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
4 | 1, 3 | sbcbid 2816 | . . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) |
5 | 4 | abbidv 2155 | . 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
6 | df-csb 2853 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
7 | df-csb 2853 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4g 2097 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 {cab 2026 [wsbc 2764 ⦋csb 2852 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-sbc 2765 df-csb 2853 |
This theorem is referenced by: csbeq2dv 2875 |
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