Theorem ralndv2 47076

Description: Second example for a theorem about a restricted universal quantification in which the restricting class depends on the bound variable: all subsets of a set are sets. (Contributed by AV, 24-Jun-2023.)

Assertion

Ref	Expression
ralndv2	⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V

Proof of Theorem ralndv2

Step	Hyp	Ref	Expression
1		vex 3467	. 2 ⊢ 𝑥 ∈ V
2	1	rgenw 3054	1 ⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V

Syntax hints:   ∈ wcel 2107  ∀wral 3050  Vcvv 3463  𝒫 cpw 4580

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 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-v 3465

This theorem is referenced by: (None)