Theorem sseq12i 3185

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 3182	. 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 1353   C_ wss 3131

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  3sstr3i  3197  3sstr4i  3198  3sstr3g  3199  3sstr4g  3200  ss2rab  3233