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Mirrors > Home > MPE Home > Th. List > cbviotavw | Structured version Visualization version GIF version |
Description: Change bound variables in a description binder. Version of cbviotav 6526 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.) |
Ref | Expression |
---|---|
cbviotavw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbviotavw | ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviotavw.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvabv 2810 | . . . . 5 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
3 | 2 | eqeq1i 2740 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜓} = {𝑧}) |
4 | 3 | abbii 2807 | . . 3 ⊢ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} |
5 | 4 | unieqi 4924 | . 2 ⊢ ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} |
6 | df-iota 6516 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
7 | df-iota 6516 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} | |
8 | 5, 6, 7 | 3eqtr4i 2773 | 1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 {cab 2712 {csn 4631 ∪ cuni 4912 ℩cio 6514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-iota 6516 |
This theorem is referenced by: cbvriotavw 7398 oeeui 8639 nosupcbv 27762 noinfcbv 27777 cbvriotavw2 36219 ellimciota 45570 fourierdlem96 46158 fourierdlem97 46159 fourierdlem98 46160 fourierdlem99 46161 fourierdlem105 46167 fourierdlem106 46168 fourierdlem108 46170 fourierdlem110 46172 fourierdlem112 46174 fourierdlem113 46175 fourierdlem115 46177 funressndmafv2rn 47173 |
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