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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 2377. Use the weaker cbvriotaw 7397 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 2824 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
| 2 | sbequ12 2251 | . . . . 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 1914 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) | |
| 5 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | |
| 6 | nfs1v 2156 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
| 7 | 5, 6 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) | 
| 8 | 3, 4, 7 | cbviota 6523 | . . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) | 
| 9 | eleq1w 2824 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 10 | sbequ 2083 | . . . . . 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 2507 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| 14 | 10, 13 | bitrdi 287 | . . . . 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 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 | |
| 17 | cbvriota.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
| 18 | 17 | nfsb 2528 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 | 
| 19 | 16, 18 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) | 
| 20 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
| 21 | 15, 19, 20 | cbviota 6523 | . . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | 
| 22 | 8, 21 | eqtri 2765 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | 
| 23 | df-riota 7388 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 24 | df-riota 7388 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
| 25 | 22, 23, 24 | 3eqtr4i 2775 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 ℩cio 6512 ℩crio 7387 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-sn 4627 df-uni 4908 df-iota 6514 df-riota 7388 | 
| This theorem is referenced by: cbvriotav 7402 | 
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