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Theorem sseq12d 2968
 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 2966 . 2 (φ → (A𝐶B𝐶))
3 sseq12d.2 . . 3 (φ𝐶 = 𝐷)
43sseq2d 2967 . 2 (φ → (B𝐶B𝐷))
52, 4bitrd 177 1 (φ → (A𝐶B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925 This theorem is referenced by:  3sstr3d  2981  3sstr4d  2982  ssdifeq0  3299  relcnvtr  4783  rdgisucinc  5912  nnaword  6020  nnawordi  6024
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