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Theorem 3eltr3d 2136
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3d.1 (𝜑𝐴𝐵)
3eltr3d.2 (𝜑𝐴 = 𝐶)
3eltr3d.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
3eltr3d (𝜑𝐶𝐷)

Proof of Theorem 3eltr3d
StepHypRef Expression
1 3eltr3d.2 . 2 (𝜑𝐴 = 𝐶)
2 3eltr3d.1 . . 3 (𝜑𝐴𝐵)
3 3eltr3d.3 . . 3 (𝜑𝐵 = 𝐷)
42, 3eleqtrd 2132 . 2 (𝜑𝐴𝐷)
51, 4eqeltrrd 2131 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  reg3exmidlemwe  4331  nnaordi  6112  icoshftf1o  8960  lincmb01cmp  8972  fzosubel  9152
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