Theorem 3eltr4g 2845

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 2832	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr4g.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	eleqtrrdi 2839	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-clel 2803

This theorem is referenced by:  riotacl2  7322  rankelun  9768  rankelpr  9769  rankelop  9770  cdivcncf  24812  rrx0el  25296  itg1addlem4  25598  cxpcn3  26656  bposlem4  27196  nosepdm  27594  mirauto  28633  ldgenpisyslem1  34146  weiunfrlem  36458  relowlpssretop  37358  0prjspnlem  42616  mapfzcons  42709  fourierdlem62  46169  fourierdlem63  46170  gpgprismgr4cycllem8  48106  line2x  48759  line2y  48760