Theorem bj-abv 32901

Description: The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-abv	|- ( A. x ph -> { x | ph } = _V )

Proof of Theorem bj-abv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1839	. . 3 |- ( A. x ph -> A. y A. x ph )
2		bj-vexwvt 32856	. . 3 |- ( A. x ph -> y e. { x | ph } )
3	1, 2	alrimih 1751	. 2 |- ( A. x ph -> A. y y e. { x | ph } )
4		eqv 3205	. 2 |- ( { x | ph } = _V <-> A. y y e. { x | ph } )
5	3, 4	sylibr 224	1 |- ( A. x ph -> { x | ph } = _V )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by: (None)