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Theorem 3eltr4d 2224
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4d.1 (𝜑𝐴𝐵)
3eltr4d.2 (𝜑𝐶 = 𝐴)
3eltr4d.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3eltr4d (𝜑𝐶𝐷)

Proof of Theorem 3eltr4d
StepHypRef Expression
1 3eltr4d.2 . 2 (𝜑𝐶 = 𝐴)
2 3eltr4d.1 . . 3 (𝜑𝐴𝐵)
3 3eltr4d.3 . . 3 (𝜑𝐷 = 𝐵)
42, 3eleqtrrd 2220 . 2 (𝜑𝐴𝐷)
51, 4eqeltrd 2217 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136
This theorem is referenced by:  ovmpodxf  5902  nnaordi  6410  iccf1o  9815  ennnfonelemrn  11961  ctiunctlemfo  11981
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