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Theorem syl5sseq 3021
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 2995 . . 3 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
43biimpa 284 . 2 ((𝐴 = 𝐶𝐵𝐴) → 𝐵𝐶)
51, 2, 4sylancl 398 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959
This theorem is referenced by:  fndmdif  5300  fneqeql2  5304  fconst4m  5409  f1opw2  5734  ecss  6178  fopwdom  6341  phplem2  6347  nn0supp  8291  monoord2  9400
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