ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sseq12d GIF version

Theorem sseq12d 3186
Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
Hypotheses
Ref Expression
sseq1d.1 (𝜑𝐴 = 𝐵)
sseq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
sseq12d (𝜑 → (𝐴𝐶𝐵𝐷))

Proof of Theorem sseq12d
StepHypRef Expression
1 sseq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21sseq1d 3184 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 sseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43sseq2d 3185 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 188 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wss 3129
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 3135  df-ss 3142
This theorem is referenced by:  3sstr3d  3199  3sstr4d  3200  ssdifeq0  3505  relcnvtr  5148  rdgisucinc  6385  oawordriexmid  6470  nnaword  6511  nnawordi  6515  sbthlem2  6956  isbth  6965  nninff  7120  infnninf  7121  infnninfOLD  7122  nnnninf  7123  nnnninfeq  7125  nnnninfeq2  7126  nninfwlpoimlemg  7172  ennnfonelemkh  12412  ennnfonelemrnh  12416  isstruct2im  12471  isstruct2r  12472  basis1  13517  baspartn  13520  eltg  13522  metss  13964  0nninf  14723  nnsf  14724  peano4nninf  14725  nninfalllem1  14727  nninfself  14732
  Copyright terms: Public domain W3C validator