Theorem ralndv2 48316

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 3457	. 2 ⊢ 𝑥 ∈ V
2	1	rgenw 3083	1 ⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V

This proof depends on syntax axioms:   ∈ wcel 2146  ∀wral 3079  Vcvv 3453  𝒫 cpw 4559

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3455

This theorem is used by: (None)