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	syl5eqelr 2844	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	syl6eleq 2849	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756

This theorem is referenced by:  rankelpr  8909  isf34lem7  9393  rmulccn  30283  xrge0mulc1cn  30296  esumpfinvallem  30445  fourierdlem62  40888