Theorem abeq1 2733

Description: Equality of a class variable and a class abstraction. Commuted form of abeq2 2732. (Contributed by NM, 20-Aug-1993.)

Assertion

Ref	Expression
abeq1	|- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem abeq1

Step	Hyp	Ref	Expression
1		abeq2 2732	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		eqcom 2629	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
3		bicom 212	. . 3 |- ( ( ph <-> x e. A ) <-> ( x e. A <-> ph ) )
4	3	albii 1747	. 2 |- ( A. x ( ph <-> x e. A ) <-> A. x ( x e. A <-> ph ) )
5	1, 2, 4	3bitr4i 292	1 |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )

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

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

This theorem is referenced by:  disj  4017  euabsn2  4260  dm0rn0  5342  dffo3  6374  dfsup2  8350  rankf  8657  dfon3  31999  dfiota3  32030  dffo3f  39364