Theorem 3eltr3g 2855

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 2844	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eleqtrdi 2849	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-clel 2817

This theorem is referenced by:  rankelpr  9562  isf34lem7  10066  rmulccn  31780  xrge0mulc1cn  31793  esumpfinvallem  31942  fourierdlem62  43599