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Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaexaiotaiota | Structured version Visualization version GIF version |
Description: The alternate iota over a wff 𝜑 is a set iff the iota and the alternate iota over 𝜑 are equal. (Contributed by AV, 25-Aug-2022.) |
Ref | Expression |
---|---|
aiotaexaiotaiota | ⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aiotaexb 43374 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V) | |
2 | reuaiotaiota 43373 | . 2 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
3 | 1, 2 | bitr3i 279 | 1 ⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃!weu 2652 Vcvv 3481 ℩cio 6293 ℩'caiota 43368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-sn 4549 df-pr 4551 df-uni 4820 df-int 4858 df-iota 6295 df-aiota 43370 |
This theorem is referenced by: (None) |
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