Theorem ssrd 3232

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 1570	. 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 3218	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
7	3, 6	sylibr 134	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   A.wal 1395   F/wnf 1508   e. wcel 2202   F/_wnfc 2361   C_ wss 3200

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213

This theorem is referenced by:  eqrd  3245  exmidomni  7340