Theorem rabss2 2100

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		pm3.45 560	. . . 4 |- ((x e. A -> x e. B) -> ((x e. A /\ ph) -> (x e. B /\ ph)))
2	1	19.20i 968	. . 3 |- (A.x(x e. A -> x e. B) -> A.x((x e. A /\ ph) -> (x e. B /\ ph)))
3		ss2ab 2087	. . 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 2029	. 2 |- (A (_ B <-> A.x(x e. A -> x e. B))
6		df-rab 1628	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
7		df-rab 1628	. . 3 |- {x e. B | ph} = {x | (x e. B /\ ph)}
8	6, 7	sseq12i 2058	. 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 950   e. wcel 1105  {cab 1440  {crab 1624   (_ wss 2018

This theorem is referenced by:  shatomistic 10410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-rab 1628  df-in 2022  df-ss 2024

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