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| Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaval | Structured version Visualization version GIF version | ||
| Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.) | 
| Ref | Expression | 
|---|---|
| aiotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eusnsn 47038 | . . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦} | |
| 2 | eqcom 2744 | . . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦}) | |
| 3 | 2 | eubii 2585 | . . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦}) | 
| 4 | 1, 3 | mpbir 231 | . . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧} | 
| 5 | eqeq1 2741 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧})) | |
| 6 | 5 | eubidv 2586 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧})) | 
| 7 | 4, 6 | mpbiri 258 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∃!𝑧{𝑥 ∣ 𝜑} = {𝑧}) | 
| 8 | absn 4645 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 9 | reuabaiotaiota 47099 | . . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
| 10 | eqcom 2744 | . . . 4 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) | |
| 11 | 9, 10 | bitri 275 | . . 3 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) | 
| 12 | 7, 8, 11 | 3imtr3i 291 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑)) | 
| 13 | iotaval 6532 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
| 14 | 12, 13 | eqtrd 2777 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃!weu 2568 {cab 2714 {csn 4626 ℩cio 6512 ℩'caiota 47095 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-iota 6514 df-aiota 47097 | 
| This theorem is referenced by: aiota0def 47108 | 
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