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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 6523 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 2811 | . . . . 5 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | 
| 3 | 2 | eqeq1i 2741 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜓} = {𝑧}) | 
| 4 | 3 | abbii 2808 | . . 3 ⊢ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} | 
| 5 | 4 | unieqi 4918 | . 2 ⊢ ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} | 
| 6 | df-iota 6513 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
| 7 | df-iota 6513 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜓} = {𝑧}} | |
| 8 | 5, 6, 7 | 3eqtr4i 2774 | 1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 {cab 2713 {csn 4625 ∪ cuni 4906 ℩cio 6511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: cbvriotavw 7399 oeeui 8641 nosupcbv 27748 noinfcbv 27763 cbvriotavw2 36238 ellimciota 45634 fourierdlem96 46222 fourierdlem97 46223 fourierdlem98 46224 fourierdlem99 46225 fourierdlem105 46231 fourierdlem106 46232 fourierdlem108 46234 fourierdlem110 46236 fourierdlem112 46238 fourierdlem113 46239 fourierdlem115 46241 funressndmafv2rn 47240 | 
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