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Mirrors > Home > ILE Home > Th. List > csbeq2d | GIF version |
Description: Formula-building deduction 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 2187 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
4 | 1, 3 | sbcbid 2938 | . . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) |
5 | 4 | abbidv 2235 | . 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
6 | df-csb 2976 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
7 | df-csb 2976 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4g 2175 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 Ⅎwnf 1421 ∈ wcel 1465 {cab 2103 [wsbc 2882 ⦋csb 2975 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-sbc 2883 df-csb 2976 |
This theorem is referenced by: csbeq2dv 2998 |
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