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Mirrors > Home > ILE Home > Th. List > csbid | GIF version |
Description: Analog of sbid 1657 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbid | ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 2853 | . 2 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} | |
2 | sbcid 2779 | . . 3 ⊢ ([𝑥 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
3 | 2 | abbii 2153 | . 2 ⊢ {𝑦 ∣ [𝑥 / 𝑥]𝑦 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
4 | abid2 2158 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
5 | 1, 3, 4 | 3eqtri 2064 | 1 ⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ 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: csbeq1a 2860 fvmpt2 5254 |
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