Theorem ninjust 3211

Description: Soundness theorem for df-nin 3212. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
ninjust	⊢ {x ∣ (x ∈ A ⊼ x ∈ B)} = {y ∣ (y ∈ A ⊼ y ∈ B)}

Distinct variable groups:   x,A,y   x,B,y

Proof of Theorem ninjust

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
2		eleq1 2413	. . 3 ⊢ (x = y → (x ∈ B ↔ y ∈ B))
3	1, 2	nanbi12d 1303	. 2 ⊢ (x = y → ((x ∈ A ⊼ x ∈ B) ↔ (y ∈ A ⊼ y ∈ B)))
4	3	cbvabv 2473	1 ⊢ {x ∣ (x ∈ A ⊼ x ∈ B)} = {y ∣ (y ∈ A ⊼ y ∈ B)}

Syntax hints:   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710  {cab 2339

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)