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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfaiota2 | Structured version Visualization version GIF version |
Description: Alternate definition of the alternate version of Russell's definition of a description binder. Definition 8.18 in [Quine] p. 56. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
dfaiota2 | ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aiota 44634 | . 2 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
2 | absn 4583 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
3 | 2 | abbii 2806 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} |
4 | 3 | inteqi 4890 | . 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 205 ∀wal 1537 = wceq 1539 {cab 2713 {csn 4565 ∩ cint 4886 ℩'caiota 44632 |
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-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-ral 3063 df-sn 4566 df-int 4887 df-aiota 44634 |
This theorem is referenced by: (None) |
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