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Theorem sseq12d 3188
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 3186 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 sseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43sseq2d 3187 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 188 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  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:  3sstr3d  3201  3sstr4d  3202  ssdifeq0  3507  relcnvtr  5150  rdgisucinc  6388  oawordriexmid  6473  nnaword  6514  nnawordi  6518  sbthlem2  6959  isbth  6968  nninff  7123  infnninf  7124  infnninfOLD  7125  nnnninf  7126  nnnninfeq  7128  nnnninfeq2  7129  nninfwlpoimlemg  7175  ennnfonelemkh  12415  ennnfonelemrnh  12419  isstruct2im  12474  isstruct2r  12475  basis1  13586  baspartn  13589  eltg  13591  metss  14033  0nninf  14792  nnsf  14793  peano4nninf  14794  nninfalllem1  14796  nninfself  14801
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