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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 6980 | . 2 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) | |
2 | vex 3439 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 531 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑)) |
4 | 3 | iotabii 6214 | . 2 ⊢ (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑)) |
5 | 1, 4 | eqtr4i 2821 | 1 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∈ wcel 2080 Vcvv 3436 ℩cio 6190 ℩crio 6979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-ext 2768 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1763 df-sb 2042 df-clab 2775 df-cleq 2787 df-clel 2862 df-rex 3110 df-v 3438 df-uni 4748 df-iota 6192 df-riota 6980 |
This theorem is referenced by: (None) |
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