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Mirrors > Home > MPE Home > Th. List > cbvriotaw | Structured version Visualization version GIF version |
Description: Change bound variable in a restricted description binder. Version of cbvriota 7381 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2369. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
cbvriotaw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvriotaw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvriotaw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvriotaw | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2814 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
2 | sbequ12 2241 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | 1, 2 | anbi12d 629 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))) |
4 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) | |
5 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | |
6 | nfs1v 2151 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
7 | 5, 6 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
8 | 3, 4, 7 | cbviotaw 6501 | . . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
9 | eleq1w 2814 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | cbvriotaw.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
11 | cbvriotaw.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | sbhypf 3537 | . . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
13 | 9, 12 | anbi12d 629 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
14 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
15 | cbvriotaw.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
16 | 15 | nfsbv 2321 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
17 | 14, 16 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
18 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
19 | 13, 17, 18 | cbviotaw 6501 | . . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
20 | 8, 19 | eqtri 2758 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
21 | df-riota 7367 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
22 | df-riota 7367 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
23 | 20, 21, 22 | 3eqtr4i 2768 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 Ⅎwnf 1783 [wsb 2065 ∈ wcel 2104 ℩cio 6492 ℩crio 7366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3954 df-ss 3964 df-sn 4628 df-uni 4908 df-iota 6494 df-riota 7367 |
This theorem is referenced by: cbvriotavwOLD 7378 disjinfi 44189 |
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