Theorem ss2ab 1551

Description: Equivalence of abstraction subclass and implication.

Assertion

Ref	Expression
ss2ab	|- ({x | ph} (_ {x | ps} <-> A.x(ph -> ps))

Proof of Theorem ss2ab

Step	Hyp	Ref	Expression
1		hbab1 1095	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
2		hbab1 1095	. . 3 |- (y e. {x | ps} -> A.x y e. {x | ps})
3	1, 2	dfss2f 1499	. 2 |- ({x | ph} (_ {x | ps} <-> A.x(x e. {x | ph} -> x e. {x | ps}))
4		abid 1094	. . . 4 |- (x e. {x | ph} <-> ph)
5		abid 1094	. . . 4 |- (x e. {x | ps} <-> ps)
6	4, 5	imbi12i 163	. . 3 |- ((x e. {x | ph} -> x e. {x | ps}) <-> (ph -> ps))
7	6	bial 695	. 2 |- (A.x(x e. {x | ph} -> x e. {x | ps}) <-> A.x(ph -> ps))
8	3, 7	bitr 151	1 |- ({x | ph} (_ {x | ps} <-> A.x(ph -> ps))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  {cab 1090   e. wcel 1092   (_ wss 1487

This theorem is referenced by:  ss2abi 1552  ss2rab 1553  uniss 1936  iunss1 2002  ssopab2 2119  mapss 3270  cfub 3703  cflim 3704  cflecard 3707

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

metamath.org