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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 7323 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 2821 | . . . 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 6454 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
5 | df-riota 7308 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | df-riota 7308 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4i 2776 | 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 6444 ℩crio 7307 |
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 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3446 df-in 3916 df-ss 3926 df-uni 4865 df-iota 6446 df-riota 7308 |
This theorem is referenced by: ordtypecbv 9412 fin23lem27 10223 zorn2g 10398 nosupcbv 27002 noinfcbv 27017 uspgredg2v 28001 usgredg2v 28004 cnlnadji 30847 nmopadjlei 30859 cvmliftlem15 33696 cvmliftiota 33699 cvmlift2 33714 cvmlift3lem7 33723 cvmlift3 33726 lshpkrlem3 37506 cdleme40v 38864 lcfl7N 39896 lcf1o 39946 lcfrlem39 39976 hdmap1cbv 40197 wessf1ornlem 43302 fourierdlem103 44345 fourierdlem104 44346 |
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