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Theorem syl5sseq 3319
 Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
syl5sseq.1 B A
syl5sseq.2 (φA = C)
Assertion
Ref Expression
syl5sseq (φB C)

Proof of Theorem syl5sseq
StepHypRef Expression
1 syl5sseq.2 . 2 (φA = C)
2 syl5sseq.1 . 2 B A
3 sseq2 3293 . . 3 (A = C → (B AB C))
43biimpa 470 . 2 ((A = C B A) → B C)
51, 2, 4sylancl 643 1 (φB C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  fconst4  5458  ecss  5966  sbthlem3  6205
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