Theorem ralndv2 44485

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 3426	. 2 ⊢ 𝑥 ∈ V
2	1	rgenw 3075	1 ⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V

Syntax hints:   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  𝒫 cpw 4530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424

This theorem is referenced by: (None)