Theorem ssrd 3147

Description: Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
ssrd.0	|- F/ x ph
ssrd.1	|- F/_ x A
ssrd.2	|- F/_ x B
ssrd.3	|- ( ph -> ( x e. A -> x e. B ) )

Assertion

Ref	Expression
ssrd	|- ( ph -> A C_ B )

Proof of Theorem ssrd

Step	Hyp	Ref	Expression
1		ssrd.0	. . 3 |- F/ x ph
2		ssrd.3	. . 3 |- ( ph -> ( x e. A -> x e. B ) )
3	1, 2	alrimi 1510	. 2 |- ( ph -> A. x ( x e. A -> x e. B ) )
4		ssrd.1	. . 3 |- F/_ x A
5		ssrd.2	. . 3 |- F/_ x B
6	4, 5	dfss2f 3133	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
7	3, 6	sylibr 133	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448   e. wcel 2136   F/_wnfc 2295   C_ wss 3116

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129

This theorem is referenced by:  eqrd  3160  exmidomni  7106