Theorem sseq12d 3214

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 3212	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3213	. 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  3227  3sstr4d  3228  ssdifeq0  3533  relcnvtr  5189  rdgisucinc  6443  oawordriexmid  6528  nnaword  6569  nnawordi  6573  sbthlem2  7024  isbth  7033  nninff  7188  nninfninc  7189  infnninf  7190  infnninfOLD  7191  nnnninf  7192  nnnninfeq  7194  nnnninfeq2  7195  nninfwlpoimlemg  7241  ennnfonelemkh  12629  ennnfonelemrnh  12633  isstruct2im  12688  isstruct2r  12689  basis1  14283  baspartn  14286  eltg  14288  metss  14730  0nninf  15648  nnsf  15649  peano4nninf  15650  nninfalllem1  15652  nninfself  15657