Theorem abbi 2463

Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
abbi	|- (A.x(ph <-> ps) <-> {x | ph} = {x | ps})

Proof of Theorem abbi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2347	. 2 |- ({x | ph} = {x | ps} <-> A.y(y e. {x | ph} <-> y e. {x | ps}))
2		nfsab1 2343	. . . 4 |- F/x y e. {x | ph}
3		nfsab1 2343	. . . 4 |- F/x y e. {x | ps}
4	2, 3	nfbi 1834	. . 3 |- F/x(y e. {x | ph} <-> y e. {x | ps})
5		nfv 1619	. . 3 |- F/y(ph <-> ps)
6		df-clab 2340	. . . . 5 |- (y e. {x | ph} <-> [y / x]ph)
7		sbequ12r 1920	. . . . 5 |- (y = x -> ([y / x]ph <-> ph))
8	6, 7	syl5bb 248	. . . 4 |- (y = x -> (y e. {x | ph} <-> ph))
9		df-clab 2340	. . . . 5 |- (y e. {x | ps} <-> [y / x]ps)
10		sbequ12r 1920	. . . . 5 |- (y = x -> ([y / x]ps <-> ps))
11	9, 10	syl5bb 248	. . . 4 |- (y = x -> (y e. {x | ps} <-> ps))
12	8, 11	bibi12d 312	. . 3 |- (y = x -> ((y e. {x | ph} <-> y e. {x | ps}) <-> (ph <-> ps)))
13	4, 5, 12	cbval 1984	. 2 |- (A.y(y e. {x | ph} <-> y e. {x | ps}) <-> A.x(ph <-> ps))
14	1, 13	bitr2i 241	1 |- (A.x(ph <-> ps) <-> {x | ph} = {x | ps})

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642  [wsb 1648   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

This theorem is referenced by:  abbii  2465  abbid  2466  rabbi  2789  dfeu2  4333  dfiota2  4340  iotabi  4348  uniabio  4349  iotanul  4354