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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 3883 | . . . 4 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑)) | |
2 | dfsbcq2 3772 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | iotabidv 6332 | . . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | eqeq12d 2834 | . . 3 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))) |
5 | vex 3495 | . . . 4 ⊢ 𝑧 ∈ V | |
6 | nfs1v 2264 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
7 | 6 | nfiotaw 6311 | . . . 4 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑) |
8 | sbequ12 2243 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 8 | iotabidv 6332 | . . . 4 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)) |
10 | 5, 7, 9 | csbief 3914 | . . 3 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) |
11 | 4, 10 | vtoclg 3565 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
12 | csbprc 4355 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = ∅) | |
13 | sbcex 3779 | . . . . . 6 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
14 | 13 | con3i 157 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝜑) |
15 | 14 | nexdv 1928 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑦[𝐴 / 𝑥]𝜑) |
16 | euex 2655 | . . . . 5 ⊢ (∃!𝑦[𝐴 / 𝑥]𝜑 → ∃𝑦[𝐴 / 𝑥]𝜑) | |
17 | 16 | con3i 157 | . . . 4 ⊢ (¬ ∃𝑦[𝐴 / 𝑥]𝜑 → ¬ ∃!𝑦[𝐴 / 𝑥]𝜑) |
18 | iotanul 6326 | . . . 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 2856 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)) |
21 | 11, 20 | pm2.61i 183 | 1 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∃wex 1771 [wsb 2060 ∈ wcel 2105 ∃!weu 2646 Vcvv 3492 [wsbc 3769 ⦋csb 3880 ∅c0 4288 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-uni 4831 df-iota 6307 |
This theorem is referenced by: csbfv12 6706 |
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