Theorem vjust 2861

Description: Soundness justification theorem for df-v 2862. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Assertion

Ref	Expression
vjust	⊢ {x ∣ x = x} = {y ∣ y = y}

Proof of Theorem vjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1676	. . . . 5 ⊢ x = x
2	1	sbt 2033	. . . 4 ⊢ [z / x]x = x
3		equid 1676	. . . . 5 ⊢ y = y
4	3	sbt 2033	. . . 4 ⊢ [z / y]y = y
5	2, 4	2th 230	. . 3 ⊢ ([z / x]x = x ↔ [z / y]y = y)
6		df-clab 2340	. . 3 ⊢ (z ∈ {x ∣ x = x} ↔ [z / x]x = x)
7		df-clab 2340	. . 3 ⊢ (z ∈ {y ∣ y = y} ↔ [z / y]y = y)
8	5, 6, 7	3bitr4i 268	. 2 ⊢ (z ∈ {x ∣ x = x} ↔ z ∈ {y ∣ y = y})
9	8	eqriv 2350	1 ⊢ {x ∣ x = x} = {y ∣ y = y}

Syntax hints:   = wceq 1642  [wsb 1648   ∈ 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by: (None)