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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 6505 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 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 2803 | . . . . 5 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
3 | 2 | eqeq1i 2735 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜓} = {𝑧}) |
4 | 3 | abbii 2800 | . . 3 ⊢ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} |
5 | 4 | unieqi 4920 | . 2 ⊢ ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} |
6 | df-iota 6494 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
7 | df-iota 6494 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} | |
8 | 5, 6, 7 | 3eqtr4i 2768 | 1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 {cab 2707 {csn 4627 ∪ cuni 4907 ℩cio 6492 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3954 df-ss 3964 df-uni 4908 df-iota 6494 |
This theorem is referenced by: cbvriotavw 7377 oeeui 8604 nosupcbv 27441 noinfcbv 27456 ellimciota 44628 fourierdlem96 45216 fourierdlem97 45217 fourierdlem98 45218 fourierdlem99 45219 fourierdlem105 45225 fourierdlem106 45226 fourierdlem108 45228 fourierdlem110 45230 fourierdlem112 45232 fourierdlem113 45233 fourierdlem115 45235 funressndmafv2rn 46229 |
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