Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > csbiota | Structured version Visualization version GIF version |
Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.) |
Ref | Expression |
---|---|
csbiota | ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3536 | . . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑)) | |
2 | dfsbcq2 3438 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | iotabidv 5872 | . . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | eqeq12d 2637 | . . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))) |
5 | vex 3203 | . . . 4 ⊢ 𝑧 ∈ V | |
6 | nfs1v 2437 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
7 | 6 | nfiota 5855 | . . . 4 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑) |
8 | sbequ12 2111 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 8 | iotabidv 5872 | . . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)) |
10 | 5, 7, 9 | csbief 3558 | . . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) |
11 | 4, 10 | vtoclg 3266 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
12 | csbprc 3980 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = ∅) | |
13 | sbcex 3445 | . . . . . 6 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
14 | 13 | con3i 150 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑) |
15 | 14 | nexdv 1864 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑) |
16 | euex 2494 | . . . . 5 ⊢ (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑) | |
17 | 16 | con3i 150 | . . . 4 ⊢ (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑) |
18 | iotanul 5866 | . . . 4 ⊢ (¬ ∃!𝑦[𝐴 / 𝑥]𝜑 → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅) | |
19 | 15, 17, 18 | 3syl 18 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝜑) = ∅) |
20 | 12, 19 | eqtr4d 2659 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
21 | 11, 20 | pm2.61i 176 | 1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∃wex 1704 [wsb 1880 ∈ wcel 1990 ∃!weu 2470 Vcvv 3200 [wsbc 3435 ⦋csb 3533 ∅c0 3915 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-uni 4437 df-iota 5851 |
This theorem is referenced by: csbfv12 6231 |
Copyright terms: Public domain | W3C validator |