Theorem bnj1241 30878

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1241.1	|- ( ph -> A C_ B )
bnj1241.2	|- ( ps -> C = A )

Assertion

Ref	Expression
bnj1241	|- ( ( ph /\ ps ) -> C C_ B )

Proof of Theorem bnj1241

Step	Hyp	Ref	Expression
1		bnj1241.2	. . . 4 |- ( ps -> C = A )
2	1	eqcomd 2628	. . 3 |- ( ps -> A = C )
3	2	adantl 482	. 2 |- ( ( ph /\ ps ) -> A = C )
4		bnj1241.1	. . 3 |- ( ph -> A C_ B )
5	4	adantr 481	. 2 |- ( ( ph /\ ps ) -> A C_ B )
6	3, 5	eqsstr3d 3640	1 |- ( ( ph /\ ps ) -> C C_ B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   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-in 3581  df-ss 3588

This theorem is referenced by:  bnj1245  31082  bnj1311  31092