Theorem ssrd 3107

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 1503	. 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 3093	. 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 1330   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089

This theorem is referenced by:  eqrd  3120  exmidomni  7022