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Mirrors > Home > MPE Home > Th. List > csbexg | Structured version Visualization version GIF version |
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 3812 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | abid2 2926 | . . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵 | |
3 | elex 3455 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
4 | 2, 3 | syl5eqel 2887 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) |
5 | 4 | alimi 1793 | . . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) |
6 | spsbc 3719 | . . . . . 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 2949 | . . . . . 6 ⊢ Ⅎ𝑥V | |
9 | 8 | sbcabel 3789 | . . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) |
10 | 7, 9 | sylibd 240 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) |
11 | 10 | imp 407 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V) |
12 | 1, 11 | syl5eqel 2887 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
13 | csbprc 4278 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
14 | 0ex 5102 | . . . 4 ⊢ ∅ ∈ V | |
15 | 13, 14 | syl6eqel 2891 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
16 | 15 | adantr 481 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
17 | 12, 16 | pm2.61ian 808 | 1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1520 ∈ wcel 2081 {cab 2775 Vcvv 3437 [wsbc 3706 ⦋csb 3811 ∅c0 4211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-nul 5101 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-nul 4212 |
This theorem is referenced by: csbex 5106 abfmpeld 30089 |
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