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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 7389 | . 2 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) | |
| 2 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) | 
| 4 | 3 | iotabii 6545 | . 2 ⊢ (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) | 
| 5 | 1, 4 | eqtr4i 2767 | 1 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ℩cio 6511 ℩crio 7388 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4907 df-iota 6513 df-riota 7389 | 
| This theorem is referenced by: (None) | 
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