ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl5eqssr GIF version

Theorem syl5eqssr 3053
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqssr.1 𝐵 = 𝐴
syl5eqssr.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
syl5eqssr (𝜑𝐴𝐶)

Proof of Theorem syl5eqssr
StepHypRef Expression
1 syl5eqssr.1 . . 3 𝐵 = 𝐴
21eqcomi 2087 . 2 𝐴 = 𝐵
3 syl5eqssr.2 . 2 (𝜑𝐵𝐶)
42, 3syl5eqss 3052 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995
This theorem is referenced by:  relcnvtr  4890  resasplitss  5120  fimacnvdisj  5125  fimacnv  5349  f1ompt  5373  tfr1onlemres  6019  tfrcllemres  6032
  Copyright terms: Public domain W3C validator