Theorem 3eltr3d 2433

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

Hypotheses

Ref	Expression
3eltr3d.1	⊢ (φ → A ∈ B)
3eltr3d.2	⊢ (φ → A = C)
3eltr3d.3	⊢ (φ → B = D)

Assertion

Ref	Expression
3eltr3d	⊢ (φ → C ∈ D)

Proof of Theorem 3eltr3d

Step	Hyp	Ref	Expression
1		3eltr3d.2	. 2 ⊢ (φ → A = C)
2		3eltr3d.1	. . 3 ⊢ (φ → A ∈ B)
3		3eltr3d.3	. . 3 ⊢ (φ → B = D)
4	2, 3	eleqtrd 2429	. 2 ⊢ (φ → A ∈ D)
5	1, 4	eqeltrrd 2428	1 ⊢ (φ → C ∈ D)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)