Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5eleq Structured version   Visualization version   GIF version

Theorem syl5eleq 2694
 Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleq.1 𝐴𝐵
syl5eleq.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
syl5eleq (𝜑𝐴𝐶)

Proof of Theorem syl5eleq
StepHypRef Expression
1 syl5eleq.1 . . 3 𝐴𝐵
21a1i 11 . 2 (𝜑𝐴𝐵)
3 syl5eleq.2 . 2 (𝜑𝐵 = 𝐶)
42, 3eleqtrd 2690 1 (𝜑𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  syl5eleqr  2695  opth1  4870  opth  4871  eqelsuc  5723  tfrlem11  7371  oalimcl  7527  omlimcl  7545  frgp0  17996  txdis  21245  ordtconlem1  29298  rankeq1o  31448
 Copyright terms: Public domain W3C validator