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Theorem syl5sseq 2987
 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 2961 . . 3 (A = 𝐶 → (BAB𝐶))
43biimpa 280 . 2 ((A = 𝐶 BA) → B𝐶)
51, 2, 4sylancl 392 1 (φB𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = 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:  fndmdif  5215  fneqeql2  5219  fconst4m  5324  f1opw2  5648  ecss  6083  fopwdom  6246  nn0supp  8010
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