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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 3511 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V) | |
3 | simpr 487 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑) | |
4 | iota2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | adantl 484 | . . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
6 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
7 | nfeu1 2668 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
8 | 6, 7 | nfan 1894 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑) |
9 | nfvd 1910 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓) | |
10 | nfcvd 2976 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴) | |
11 | 2, 3, 5, 8, 9, 10 | iota2df 6335 | . 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
12 | 1, 11 | sylan 582 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∃!weu 2647 Vcvv 3493 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-v 3495 df-sbc 3771 df-un 3939 df-sn 4560 df-pr 4562 df-uni 4831 df-iota 6307 |
This theorem is referenced by: iotan0 6338 pczpre 16176 pcdiv 16181 rngurd 30850 nosupno 33196 nosupfv 33199 bj-nuliota 34342 unirep 34980 ellimciota 41885 |
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