Theorem snjust 3741

Description: Soundness justification theorem for df-sn 3742. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
snjust	⊢ {x ∣ x = A} = {y ∣ y = A}

Distinct variable groups:   x,A   y,A

Proof of Theorem snjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . 3 ⊢ (x = z → (x = A ↔ z = A))
2	1	cbvabv 2473	. 2 ⊢ {x ∣ x = A} = {z ∣ z = A}
3		eqeq1 2359	. . 3 ⊢ (z = y → (z = A ↔ y = A))
4	3	cbvabv 2473	. 2 ⊢ {z ∣ z = A} = {y ∣ y = A}
5	2, 4	eqtri 2373	1 ⊢ {x ∣ x = A} = {y ∣ y = A}

Syntax hints:   = wceq 1642  {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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by: (None)