Theorem abeq2f 2792

Description: Equality of a class variable and a class abstraction. In this version, the fact that x is a non-free variable in A is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)

Hypothesis

Ref	Expression
abeq2f.0	|- F/_ x A

Assertion

Ref	Expression
abeq2f	|- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )

Proof of Theorem abeq2f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abeq2f.0	. . . 4 |- F/_ x A
2	1	nfcrii 2757	. . 3 |- ( y e. A -> A. x y e. A )
3		hbab1 2611	. . 3 |- ( y e. { x | ph } -> A. x y e. { x | ph } )
4	2, 3	cleqh 2724	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> x e. { x | ph } ) )
5		abid 2610	. . . 4 |- ( x e. { x | ph } <-> ph )
6	5	bibi2i 327	. . 3 |- ( ( x e. A <-> x e. { x | ph } ) <-> ( x e. A <-> ph ) )
7	6	albii 1747	. 2 |- ( A. x ( x e. A <-> x e. { x | ph } ) <-> A. x ( x e. A <-> ph ) )
8	4, 7	bitri 264	1 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751

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

This theorem is referenced by:  rabid2f  3119  mptfnf  6015