ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl6eqss GIF version

Theorem syl6eqss 2995
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
syl6eqss.1 (𝜑𝐴 = 𝐵)
syl6eqss.2 𝐵𝐶
Assertion
Ref Expression
syl6eqss (𝜑𝐴𝐶)

Proof of Theorem syl6eqss
StepHypRef Expression
1 syl6eqss.1 . 2 (𝜑𝐴 = 𝐵)
2 syl6eqss.2 . . 3 𝐵𝐶
32a1i 9 . 2 (𝜑𝐵𝐶)
41, 3eqsstrd 2979 1 (𝜑𝐴𝐶)
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:  syl6eqssr  2996  resasplitss  5069  fimacnv  5296  bj-nntrans  10049
  Copyright terms: Public domain W3C validator