Theorem rabss2 2129

Description: Subclass law for restricted abstraction.

Assertion

Ref	Expression
rabss2	|- (A (_ B -> {x e. A | ph} (_ {x e. B | ph})

Distinct variable groups:   x,A   x,B

Proof of Theorem rabss2

Step	Hyp	Ref	Expression
1		pm3.45 562	. . . 4 |- ((x e. A -> x e. B) -> ((x e. A /\ ph) -> (x e. B /\ ph)))
2	1	19.20i 992	. . 3 |- (A.x(x e. A -> x e. B) -> A.x((x e. A /\ ph) -> (x e. B /\ ph)))
3		ss2ab 2116	. . 3 |- ({x | (x e. A /\ ph)} (_ {x | (x e. B /\ ph)} <-> A.x((x e. A /\ ph) -> (x e. B /\ ph)))
4	2, 3	sylibr 200	. 2 |- (A.x(x e. A -> x e. B) -> {x | (x e. A /\ ph)} (_ {x | (x e. B /\ ph)})
5		dfss2 2058	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
6		df-rab 1652	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
7		df-rab 1652	. . 3 |- {x e. B | ph} = {x | (x e. B /\ ph)}
8	6, 7	sseq12i 2087	. 2 |- ({x e. A | ph} (_ {x e. B | ph} <-> {x | (x e. A /\ ph)} (_ {x | (x e. B /\ ph)})
9	4, 5, 8	3imtr4 219	1 |- (A (_ B -> {x e. A | ph} (_ {x e. B | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  {cab 1463  {crab 1648   (_ wss 2047

This theorem is referenced by:  shatomistic 10288

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-in 2051  df-ss 2053

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