Theorem ralss 3221

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 3149	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd 394	. . . 4 |- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	imbi1d 231	. . 3 |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( ( x e. B /\ x e. A ) -> ph ) ) )
4		impexp 263	. . 3 |- ( ( ( x e. B /\ x e. A ) -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) )
5	3, 4	bitrdi 196	. 2 |- ( A C_ B -> ( ( x e. A -> ph ) <-> ( x e. B -> ( x e. A -> ph ) ) ) )
6	5	ralbidv2 2479	1 |- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   A.wral 2455   C_ wss 3129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3135  df-ss 3142

This theorem is referenced by: (None)