Theorem ssdf 39247

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
ssdf.1	|- F/ x ph
ssdf.2	|- ( ( ph /\ x e. A ) -> x e. B )

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem ssdf

Step	Hyp	Ref	Expression
1		ssdf.1	. . 3 |- F/ x ph
2		ssdf.2	. . . 4 |- ( ( ph /\ x e. A ) -> x e. B )
3	2	ex 450	. . 3 |- ( ph -> ( x e. A -> x e. B ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A x e. B )
5		dfss3 3592	. 2 |- ( A C_ B <-> A. x e. A x e. B )
6	4, 5	sylibr 224	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   A.wral 2912   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-in 3581  df-ss 3588

This theorem is referenced by:  ssd  39252  smfaddlem2  40972  smfadd  40973  smfmullem4  41001  smfmul  41002  smflimsuplem4  41029