Theorem abbi2dv 2469

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Hypothesis

Ref	Expression
abbirdv.1	|- (ph -> (x e. A <-> ps))

Assertion

Ref	Expression
abbi2dv	|- (ph -> A = {x | ps})

Distinct variable groups:   x,A   ph,x

Allowed substitution hint:   ps(x)

Proof of Theorem abbi2dv

Step	Hyp	Ref	Expression
1		abbirdv.1	. . 3 |- (ph -> (x e. A <-> ps))
2	1	alrimiv 1631	. 2 |- (ph -> A.x(x e. A <-> ps))
3		abeq2 2459	. 2 |- (A = {x | ps} <-> A.x(x e. A <-> ps))
4	2, 3	sylibr 203	1 |- (ph -> A = {x | ps})

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by:  sbab  2476  iftrue  3669  iffalse  3670  phialllem1  4617  isoini  5498