Theorem vjust 2642

Description: Soundness justification theorem for df-v 2643. (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 1645	. . . . 5 ⊢ 𝑥 = 𝑥
2	1	sbt 1725	. . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥
3		equid 1645	. . . . 5 ⊢ 𝑦 = 𝑦
4	3	sbt 1725	. . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦
5	2, 4	2th 173	. . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6		df-clab 2087	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7		df-clab 2087	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
8	5, 6, 7	3bitr4i 211	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦})
9	8	eqriv 2097	1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Syntax hints:   = wceq 1299   ∈ wcel 1448  [wsb 1703  {cab 2086

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093

This theorem is referenced by: (None)