Theorem riotav 5883

Description: An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)

Assertion

Ref	Expression
riotav	⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)

Proof of Theorem riotav

Step	Hyp	Ref	Expression
1		df-riota 5877	. 2 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 2766	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 303	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	iotabii 5242	. 2 ⊢ (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	eqtr4i 2220	1 ⊢ (℩𝑥 ∈ V 𝜑) = (℩𝑥𝜑)

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763  ℩cio 5217  ℩crio 5876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3840  df-iota 5219  df-riota 5877

This theorem is referenced by: (None)