Theorem foo3 10652

Description: A theorem about the universal class.

Hypothesis

Ref	Expression
foo3.1	|- ph

Assertion

Ref	Expression
foo3	|- V = {x | ph}

Proof of Theorem foo3

Step	Hyp	Ref	Expression
1		df-v 1858	. 2 |- V = {x | x = x}
2		equid 1162	. . . 4 |- x = x
3		foo3.1	. . . 4 |- ph
4	2, 3	2th 723	. . 3 |- (x = x <-> ph)
5	4	abbii 1618	. 2 |- {x | x = x} = {x | ph}
6	1, 5	eqtri 1538	1 |- V = {x | ph}

Syntax hints:   = wceq 992  {cab 1505  Vcvv 1857

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858

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