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Theorem 3eltr4g 2263
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4g.1 (𝜑𝐴𝐵)
3eltr4g.2 𝐶 = 𝐴
3eltr4g.3 𝐷 = 𝐵
Assertion
Ref Expression
3eltr4g (𝜑𝐶𝐷)

Proof of Theorem 3eltr4g
StepHypRef Expression
1 3eltr4g.1 . 2 (𝜑𝐴𝐵)
2 3eltr4g.2 . . 3 𝐶 = 𝐴
3 3eltr4g.3 . . 3 𝐷 = 𝐵
42, 3eleq12i 2245 . 2 (𝐶𝐷𝐴𝐵)
51, 4sylibr 134 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:  riotacl2  5838  2strop1g  12561
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