Theorem sseq12i 3207

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
sseq1i.1	|- A = B
sseq12i.2	|- C = D

Assertion

Ref	Expression
sseq12i	|- ( A C_ C <-> B C_ D )

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 |- A = B
2		sseq12i.2	. 2 |- C = D
3		sseq12 3204	. 2 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )
4	1, 2, 3	mp2an 426	1 |- ( A C_ C <-> B C_ D )

Syntax hints:   <-> wb 105   = wceq 1364   C_ wss 3153

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  3sstr3i  3219  3sstr4i  3220  3sstr3g  3221  3sstr4g  3222  ss2rab  3255