Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sseqtr4d GIF version

Theorem sseqtr4d 2982
 Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtr4d.1 (𝜑𝐴𝐵)
sseqtr4d.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
sseqtr4d (𝜑𝐴𝐶)

Proof of Theorem sseqtr4d
StepHypRef Expression
1 sseqtr4d.1 . 2 (𝜑𝐴𝐵)
2 sseqtr4d.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2045 . 2 (𝜑𝐵 = 𝐶)
41, 3sseqtrd 2981 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:  syl5sseqr  2994  fnfvima  5393  tfrlemiubacc  5944  rdgivallem  5968
 Copyright terms: Public domain W3C validator