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Mirrors > Home > MPE Home > Th. List > dfiota2 | Structured version Visualization version GIF version |
Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Ref | Expression |
---|---|
dfiota2 | ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iota 6308 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
2 | absn 4577 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
3 | 2 | abbii 2886 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
4 | 3 | unieqi 4841 | . 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
5 | 1, 4 | eqtri 2844 | 1 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 = wceq 1528 {cab 2799 {csn 4559 ∪ cuni 4832 ℩cio 6306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-rex 3144 df-sn 4560 df-uni 4833 df-iota 6308 |
This theorem is referenced by: nfiota1 6310 nfiotadw 6311 nfiotad 6313 cbviotaw 6315 cbviota 6317 sb8iota 6319 iotaval 6323 iotanul 6327 fv2 6659 |
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