Theorem 3eltr3g 2880

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 2869	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eleqtrdi 2874	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-clel 2839

This theorem is used by:  rankelpr  9914  isf34lem7  10481  rmulccn  34627  xrge0mulc1cn  34640  esumpfinvallem  34773  fourierdlem62  47381