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Mirrors > Home > MPE Home > Th. List > cbvriota | Structured version Visualization version GIF version |
Description: Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbvriotaw 7123 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvriota.1 | ⊢ Ⅎ𝑦𝜑 |
cbvriota.2 | ⊢ Ⅎ𝑥𝜓 |
cbvriota.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvriota | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2895 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
2 | sbequ12 2253 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | 1, 2 | anbi12d 632 | . . . 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 2160 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
7 | 5, 6 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
8 | 3, 4, 7 | cbviota 6323 | . . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
9 | eleq1w 2895 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | sbequ 2090 | . . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
11 | cbvriota.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝜓 | |
12 | cbvriota.3 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
13 | 11, 12 | sbie 2544 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
14 | 10, 13 | syl6bb 289 | . . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
15 | 9, 14 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
16 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
17 | cbvriota.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
18 | 17 | nfsb 2565 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
19 | 16, 18 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
20 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
21 | 15, 19, 20 | cbviota 6323 | . . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
22 | 8, 21 | eqtri 2844 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
23 | df-riota 7114 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
24 | df-riota 7114 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
25 | 22, 23, 24 | 3eqtr4i 2854 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 [wsb 2069 ∈ wcel 2114 ℩cio 6312 ℩crio 7113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-sn 4568 df-uni 4839 df-iota 6314 df-riota 7114 |
This theorem is referenced by: cbvriotav 7128 |
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