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| Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.) | 
| Ref | Expression | 
|---|---|
| csbexg | ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-csb 3899 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
| 2 | abid2 2878 | . . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵 | |
| 3 | elex 3500 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
| 4 | 2, 3 | eqeltrid 2844 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) | 
| 5 | 4 | alimi 1810 | . . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) | 
| 6 | spsbc 3800 | . . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)) | |
| 7 | 5, 6 | syl5 34 | . . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → [𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V)) | 
| 8 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥V | |
| 9 | 8 | sbcabel 3877 | . . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) | 
| 10 | 7, 9 | sylibd 239 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) | 
| 11 | 10 | imp 406 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V) | 
| 12 | 1, 11 | eqeltrid 2844 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | 
| 13 | csbprc 4408 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
| 14 | 0ex 5306 | . . . 4 ⊢ ∅ ∈ V | |
| 15 | 13, 14 | eqeltrdi 2848 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | 
| 16 | 15 | adantr 480 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | 
| 17 | 12, 16 | pm2.61ian 811 | 1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1537 ∈ wcel 2107 {cab 2713 Vcvv 3479 [wsbc 3787 ⦋csb 3898 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: csbex 5310 abfmpeld 32665 | 
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