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Mirrors > Home > MPE Home > Th. List > iota2d | Structured version Visualization version GIF version |
Description: A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.) |
Ref | Expression |
---|---|
iota2df.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
iota2df.2 | ⊢ (𝜑 → ∃!𝑥𝜓) |
iota2df.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
iota2d | ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota2df.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
2 | iota2df.2 | . 2 ⊢ (𝜑 → ∃!𝑥𝜓) | |
3 | iota2df.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
4 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜑 | |
5 | nfvd 1918 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
6 | nfcvd 2908 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | iota2df 6420 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃!weu 2568 ℩cio 6389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-v 3434 df-un 3892 df-in 3894 df-ss 3904 df-sn 4562 df-pr 4564 df-uni 4840 df-iota 6391 |
This theorem is referenced by: erov 8603 psgnvalii 19117 q1peqb 25319 |
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