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Theorem eqsstrdi 3180
 Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
eqsstrdi.1 (𝜑𝐴 = 𝐵)
eqsstrdi.2 𝐵𝐶
Assertion
Ref Expression
eqsstrdi (𝜑𝐴𝐶)

Proof of Theorem eqsstrdi
StepHypRef Expression
1 eqsstrdi.1 . 2 (𝜑𝐴 = 𝐵)
2 eqsstrdi.2 . . 3 𝐵𝐶
32a1i 9 . 2 (𝜑𝐵𝐶)
41, 3eqsstrd 3164 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ⊆ wss 3102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115 This theorem is referenced by:  eqsstrrdi  3181  resasplitss  5346  fimacnv  5593  en2other2  7114  exmidfodomrlemim  7119  pw1on  7144  suplocexprlemex  7625  ennnfonelemkh  12113  toponsspwpwg  12380  ntrss2  12481  cnprcl2k  12566  reldvg  13008  bj-nntrans  13485  nninfsellemsuc  13546
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