Theorem 3eltr3g 2857

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
3eltr3g.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr3g.2	⊢ 𝐴 = 𝐶
3eltr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eltr3g	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr3g

Step	Hyp	Ref	Expression
1		3eltr3g.2	. . 3 ⊢ 𝐴 = 𝐶
2		3eltr3g.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	eqeltrrid 2846	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eleqtrdi 2851	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-clel 2816

This theorem is referenced by:  rankelpr  9792  isf34lem7  10296  rmulccn  34124  xrge0mulc1cn  34137  esumpfinvallem  34270  fourierdlem62  46625