Theorem 3eltr3i 2220

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr3.1	⊢ 𝐴 ∈ 𝐵
3eltr3.2	⊢ 𝐴 = 𝐶
3eltr3.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eltr3i	⊢ 𝐶 ∈ 𝐷

Proof of Theorem 3eltr3i

Step	Hyp	Ref	Expression
1		3eltr3.2	. 2 ⊢ 𝐴 = 𝐶
2		3eltr3.1	. . 3 ⊢ 𝐴 ∈ 𝐵
3		3eltr3.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	eleqtri 2214	. 2 ⊢ 𝐴 ∈ 𝐷
5	1, 4	eqeltrri 2213	1 ⊢ 𝐶 ∈ 𝐷

Syntax hints:   = wceq 1331   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135

This theorem is referenced by: (None)