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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 7379 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 2816 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | cbvriotavw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
4 | 3 | cbviotavw 6503 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
5 | df-riota 7364 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | df-riota 7364 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4i 2770 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ℩cio 6493 ℩crio 7363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-iota 6495 df-riota 7364 |
This theorem is referenced by: ordtypecbv 9511 fin23lem27 10322 zorn2g 10497 nosupcbv 27202 noinfcbv 27217 uspgredg2v 28478 usgredg2v 28481 cnlnadji 31324 nmopadjlei 31336 cvmliftlem15 34284 cvmliftiota 34287 cvmlift2 34302 cvmlift3lem7 34311 cvmlift3 34314 lshpkrlem3 37977 cdleme40v 39335 lcfl7N 40367 lcf1o 40417 lcfrlem39 40447 hdmap1cbv 40668 wessf1ornlem 43872 fourierdlem103 44915 fourierdlem104 44916 |
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