Theorem ralss 3213

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 3141	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 392	. . . 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	bitrdi 195	. 2 |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) ) )
6	5	ralbidv2 2472	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 2141   A.wral 2448   C_ wss 3121

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-in 3127  df-ss 3134

This theorem is referenced by: (None)