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Theorem syl5sseq 2993
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
syl5sseq.1 𝐵𝐴
syl5sseq.2 (𝜑𝐴 = 𝐶)
Assertion
Ref Expression
syl5sseq (𝜑𝐵𝐶)

Proof of Theorem syl5sseq
StepHypRef Expression
1 syl5sseq.2 . 2 (𝜑𝐴 = 𝐶)
2 syl5sseq.1 . 2 𝐵𝐴
3 sseq2 2967 . . 3 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
43biimpa 280 . 2 ((𝐴 = 𝐶𝐵𝐴) → 𝐵𝐶)
51, 2, 4sylancl 392 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wss 2917
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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931
This theorem is referenced by:  fndmdif  5272  fneqeql2  5276  fconst4m  5381  f1opw2  5706  ecss  6147  fopwdom  6310  phplem2  6316  nn0supp  8234  monoord2  9236
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