Theorem vjust 3431

Description: Justification theorem for df-v 3432. (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 2018	. . . 4 ⊢ 𝑥 = 𝑥
2	1	vexw 2722	. . 3 ⊢ 𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥}
3		equid 2018	. . . 4 ⊢ 𝑦 = 𝑦
4	3	vexw 2722	. . 3 ⊢ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦}
5	2, 4	2th 263	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦})
6	5	eqriv 2736	1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Syntax hints:   = wceq 1541   ∈ wcel 2109  {cab 2716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731

This theorem is referenced by: (None)