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Mirrors > Home > MPE Home > Th. List > riotav | Structured version Visualization version GIF version |
Description: An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
Ref | Expression |
---|---|
riotav | ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 7114 | . 2 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 533 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | iotabii 6340 | . 2 ⊢ (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | eqtr4i 2847 | 1 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ℩cio 6312 ℩crio 7113 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-iota 6314 df-riota 7114 |
This theorem is referenced by: (None) |
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