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| 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 6513 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
| 2 | absn 4644 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 3 | 2 | abbii 2808 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} | 
| 4 | 3 | unieqi 4918 | . 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} | 
| 5 | 1, 4 | eqtri 2764 | 1 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 = 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-sn 4626 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: nfiota1 6515 nfiotadw 6516 nfiotad 6518 cbviotaw 6520 cbviota 6522 sb8iota 6524 iotavalOLD 6534 iotanul 6538 fv2 6900 | 
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