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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 3459 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | simpl 486 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V) | |
3 | simpr 488 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑) | |
4 | iota2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | adantl 485 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
6 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
7 | nfeu1 2649 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
8 | 6, 7 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑) |
9 | nfvd 1916 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓) | |
10 | nfcvd 2956 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴) | |
11 | 2, 3, 5, 8, 9, 10 | iota2df 6311 | . 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
12 | 1, 11 | sylan 583 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃!weu 2628 Vcvv 3441 ℩cio 6281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 |
This theorem is referenced by: iotan0 6314 pczpre 16174 pcdiv 16179 rngurd 30907 nosupno 33316 nosupfv 33319 bj-nuliota 34474 unirep 35151 ellimciota 42256 |
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