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Theorem 3eltr3d 2260
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 2256 . 2 (𝜑𝐴𝐷)
51, 4eqeltrrd 2255 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173
This theorem is referenced by:  reg3exmidlemwe  4574  nnaordi  6502  icoshftf1o  9965  lincmb01cmp  9977  fzosubel  10167  cnmpt2res  13430  dvcnp2cntop  13796
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