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Theorem 3eltr3g 2225
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3g.1 (𝜑𝐴𝐵)
3eltr3g.2 𝐴 = 𝐶
3eltr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3eltr3g (𝜑𝐶𝐷)

Proof of Theorem 3eltr3g
StepHypRef Expression
1 3eltr3g.1 . 2 (𝜑𝐴𝐵)
2 3eltr3g.2 . . 3 𝐴 = 𝐶
3 3eltr3g.3 . . 3 𝐵 = 𝐷
42, 3eleq12i 2208 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 121 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: (None)
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