Theorem ssdf2 39331

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
ssdf2.p	|- F/ x ph
ssdf2.a	|- F/_ x A
ssdf2.b	|- F/_ x B
ssdf2.x	|- ( ( ph /\ x e. A ) -> x e. B )

Assertion

Ref	Expression
ssdf2	|- ( ph -> A C_ B )

Proof of Theorem ssdf2

Step	Hyp	Ref	Expression
1		ssdf2.p	. 2 |- F/ x ph
2		ssdf2.a	. 2 |- F/_ x A
3		ssdf2.b	. 2 |- F/_ x B
4		ssdf2.x	. . 3 |- ( ( ph /\ x e. A ) -> x e. B )
5	4	ex 450	. 2 |- ( ph -> ( x e. A -> x e. B ) )
6	1, 2, 3, 5	ssrd 3608	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   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-nfc 2753  df-in 3581  df-ss 3588

This theorem is referenced by:  supminfxr2  39699