Theorem ssrd 3253

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 1575	. 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 3239	. 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 1400   F/wnf 1513   e. wcel 2209   F/_wnfc 2379   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233

This theorem is referenced by:  eqrd  3266  exmidomni  7472