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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 3673 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | abid2 2881 | . . . . . . . 8 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵 | |
3 | elex 3350 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
4 | 2, 3 | syl5eqel 2841 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → {𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) |
5 | 4 | alimi 1886 | . . . . . 6 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ∀𝑥{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V) |
6 | spsbc 3587 | . . . . . 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 2900 | . . . . . 6 ⊢ Ⅎ𝑥V | |
9 | 8 | sbcabel 3656 | . . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦 ∣ 𝑦 ∈ 𝐵} ∈ V ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) |
10 | 7, 9 | sylibd 229 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 𝐵 ∈ 𝑊 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V)) |
11 | 10 | imp 444 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ V) |
12 | 1, 11 | syl5eqel 2841 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
13 | csbprc 4121 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
14 | 0ex 4940 | . . . 4 ⊢ ∅ ∈ V | |
15 | 13, 14 | syl6eqel 2845 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
16 | 15 | adantr 472 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
17 | 12, 16 | pm2.61ian 866 | 1 ⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1628 ∈ wcel 2137 {cab 2744 Vcvv 3338 [wsbc 3574 ⦋csb 3672 ∅c0 4056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-nul 4939 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2045 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-nul 4057 |
This theorem is referenced by: csbex 4943 abfmpeld 29761 |
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