Theorem ralss 3158

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
ralss	|- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem ralss

Step	Hyp	Ref	Expression
1		ssel 3086	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 391	. . . 4 |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	imbi1d 230	. . 3 |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( ( x e. B /\ x e. A ) -> ph ) ) )
4		impexp 261	. . 3 |- ( ( ( x e. B /\ x e. A ) -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) )
5	3, 4	syl6bb 195	. 2 |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) ) )
6	5	ralbidv2 2437	1 |- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   A.wral 2414   C_ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-in 3072  df-ss 3079

This theorem is referenced by: (None)