Theorem rabbirdv 2218

Description: Deduction from wff to restricted class abstraction.

Hypothesis

Ref	Expression
rabbirdv.1	|- (ph -> (x e. B -> (x e. A <-> ch)))

Assertion

Ref	Expression
rabbirdv	|- (ph -> (B i^i A) = {x e. B | ch})

Distinct variable groups:   ph,x   x,A   x,B

Proof of Theorem rabbirdv

Step	Hyp	Ref	Expression
1		rabbirdv.1	. . . . 5 |- (ph -> (x e. B -> (x e. A <-> ch)))
2	1	pm5.32d 646	. . . 4 |- (ph -> ((x e. B /\ x e. A) <-> (x e. B /\ ch)))
3		elin 2204	. . . 4 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
4	2, 3	syl5bb 531	. . 3 |- (ph -> (x e. (B i^i A) <-> (x e. B /\ ch)))
5	4	abbi2dv 1576	. 2 |- (ph -> (B i^i A) = {x | (x e. B /\ ch)})
6		df-rab 1650	. 2 |- {x e. B | ch} = {x | (x e. B /\ ch)}
7	5, 6	syl6eqr 1523	1 |- (ph -> (B i^i A) = {x e. B | ch})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462  {crab 1646   i^i cin 2043

This theorem is referenced by:  pjvect 9598  pjocvect 9599

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-rab 1650  df-v 1809  df-in 2048

Copyright terms: Public domain