Theorem birabdv 1343

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule).

Hypothesis

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

Assertion

Ref	Expression
birabdv	|- (ph -> {x e. A | ps} = {x e. A | ch})

Distinct variable group(s):   ph,x

Proof of Theorem birabdv

Step	Hyp	Ref	Expression
1		birabdv.1	. . . 4 |- (ph -> (x e. A -> (ps <-> ch)))
2	1	pm5.32d 491	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
3	2	biabdv 1183	. 2 |- (ph -> {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)})
4		df-rab 1208	. 2 |- {x e. A | ps} = {x | (x e. A /\ ps)}
5		df-rab 1208	. 2 |- {x e. A | ch} = {x | (x e. A /\ ch)}
6	3, 4, 5	3eqtr4g 1147	1 |- (ph -> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  {crab 1204

This theorem is referenced by:  birabsdv 1344  onsucmin 2323

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-rab 1208

metamath.org