Theorem 3eltr3g 2435

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

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.1	. 2 ⊢ (φ → A ∈ B)
2		3eltr3g.2	. . 3 ⊢ A = C
3		3eltr3g.3	. . 3 ⊢ B = D
4	2, 3	eleq12i 2418	. 2 ⊢ (A ∈ B ↔ C ∈ D)
5	1, 4	sylib 188	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)