Theorem ralndv2 43380

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 3494	. 2 ⊢ 𝑥 ∈ V
2	1	rgenw 3149	1 ⊢ ∀𝑥 ∈ 𝒫 𝑥𝑥 ∈ V

Syntax hints:   ∈ wcel 2113  ∀wral 3137  Vcvv 3491  𝒫 cpw 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-ral 3142  df-v 3493

This theorem is referenced by: (None)