Theorem sseq12d 3215

Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseq1d.1	|- ( ph -> A = B )
sseq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
sseq12d	|- ( ph -> ( A C_ C <-> B C_ D ) )

Proof of Theorem sseq12d

Step	Hyp	Ref	Expression
1		sseq1d.1	. . 3 |- ( ph -> A = B )
2	1	sseq1d 3213	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3214	. 2 |- ( ph -> ( B C_ C <-> B C_ D ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  3sstr3d  3228  3sstr4d  3229  ssdifeq0  3534  relcnvtr  5190  rdgisucinc  6445  oawordriexmid  6530  nnaword  6571  nnawordi  6575  sbthlem2  7026  isbth  7035  nninff  7190  nninfninc  7191  infnninf  7192  infnninfOLD  7193  nnnninf  7194  nnnninfeq  7196  nnnninfeq2  7197  nninfwlpoimlemg  7243  ennnfonelemkh  12640  ennnfonelemrnh  12644  isstruct2im  12699  isstruct2r  12700  basis1  14309  baspartn  14312  eltg  14314  metss  14756  0nninf  15677  nnsf  15678  peano4nninf  15679  nninfalllem1  15681  nninfself  15686