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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 7376 | . 2 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3475 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | iotabii 6533 | . 2 ⊢ (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | eqtr4i 2759 | 1 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ℩cio 6498 ℩crio 7375 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 df-uni 4909 df-iota 6500 df-riota 7376 |
This theorem is referenced by: (None) |
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