Theorem 3eltr4g 2886

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)

Hypotheses

Ref	Expression
3eltr4g.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr4g.2	⊢ 𝐶 = 𝐴
3eltr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3eltr4g	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr4g

Step	Hyp	Ref	Expression
1		3eltr4g.2	. . 3 ⊢ 𝐶 = 𝐴
2		3eltr4g.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	eqeltrid 2873	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr4g.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	eleqtrrdi 2880	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-clel 2844

This theorem is referenced by:  riotacl2  7381  rankelun  9840  rankelpr  9841  rankelop  9842  cdivcncf  25045  rrx0el  25522  itg1addlem4  25823  cxpcn3  26875  bposlem4  27413  nosepdm  27810  mirauto  28919  ldgenpisyslem1  34494  weiunfrlem  36860  relowlpssretop  37893  0prjspnlem  43240  mapfzcons  43332  fourierdlem62  46767  fourierdlem63  46768  gpgprismgr4cycllem8  48749  line2x  49412  line2y  49413