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| Mirrors > Home > MPE Home > Th. List > iota2 | Structured version Visualization version GIF version | ||
| Description: The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
| Ref | Expression |
|---|---|
| iota2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| iota2 | ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
| 2 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V) | |
| 3 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑) | |
| 4 | iota2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 5 | 4 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
| 6 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
| 7 | nfeu1 2588 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
| 8 | 6, 7 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑) |
| 9 | nfvd 1915 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓) | |
| 10 | nfcvd 2906 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴) | |
| 11 | 2, 3, 5, 8, 9, 10 | iota2df 6548 | . 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
| 12 | 1, 11 | sylan 580 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 ℩cio 6512 |
| 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-3an 1089 df-tru 1543 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-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: iotan0 6551 pczpre 16885 pcdiv 16890 ringurd 20182 nosupno 27748 nosupfv 27751 noinfno 27763 noinffv 27766 bj-nuliota 37058 unirep 37721 ellimciota 45629 |
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