Theorem 3eltr4g 2930

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 2917	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr4g.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	eleqtrrdi 2924	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  riotacl2  7116  rankelun  9287  rankelpr  9288  rankelop  9289  cdivcncf  23508  rrx0el  23984  itg1addlem4  24283  cxpcn3  25315  bposlem4  25849  mirauto  26456  ldgenpisyslem1  31429  nosepdm  33195  relowlpssretop  34661  0prjspnlem  39360  mapfzcons  39405  fourierdlem62  42543  fourierdlem63  42544  line2x  44826  line2y  44827