Theorem 3eltr4g 2856

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 2843	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr4g.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	eleqtrrdi 2850	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816

This theorem is referenced by:  riotacl2  7241  rankelun  9640  rankelpr  9641  rankelop  9642  cdivcncf  24094  rrx0el  24572  itg1addlem4  24873  itg1addlem4OLD  24874  cxpcn3  25911  bposlem4  26445  mirauto  27055  ldgenpisyslem1  32139  nosepdm  33895  relowlpssretop  35543  0prjspnlem  40468  mapfzcons  40546  fourierdlem62  43690  fourierdlem63  43691  line2x  46078  line2y  46079