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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 5889 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
2 | df-sn 4211 | . . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
3 | 2 | eqeq2i 2663 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) |
4 | abbi 2766 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
5 | 3, 4 | bitr4i 267 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
6 | 5 | abbii 2768 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
7 | 6 | unieqi 4477 | . 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
8 | 1, 7 | eqtri 2673 | 1 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1521 = wceq 1523 {cab 2637 {csn 4210 ∪ cuni 4468 ℩cio 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rex 2947 df-sn 4211 df-uni 4469 df-iota 5889 |
This theorem is referenced by: nfiota1 5891 nfiotad 5892 cbviota 5894 sb8iota 5896 iotaval 5900 iotanul 5904 fv2 6224 |
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