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Mirrors > Home > MPE Home > Th. List > cbvriotavw | Structured version Visualization version GIF version |
Description: Change bound variable in a restricted description binder. Version of cbvriotav 7380 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.) |
Ref | Expression |
---|---|
cbvriotavw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvriotavw | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2817 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | cbvriotavw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
4 | 3 | cbviotavw 6504 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
5 | df-riota 7365 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | df-riota 7365 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4i 2771 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ℩cio 6494 ℩crio 7364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-uni 4910 df-iota 6496 df-riota 7365 |
This theorem is referenced by: ordtypecbv 9512 fin23lem27 10323 zorn2g 10498 nosupcbv 27205 noinfcbv 27220 uspgredg2v 28481 usgredg2v 28484 cnlnadji 31329 nmopadjlei 31341 cvmliftlem15 34289 cvmliftiota 34292 cvmlift2 34307 cvmlift3lem7 34316 cvmlift3 34319 lshpkrlem3 37982 cdleme40v 39340 lcfl7N 40372 lcf1o 40422 lcfrlem39 40452 hdmap1cbv 40673 wessf1ornlem 43882 fourierdlem103 44925 fourierdlem104 44926 |
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