Theorem riotav 6985

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 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)

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)