Theorem eujust 1899

Description: A soundness justification theorem for df-eu 1900, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
eujust	⊢ (∃y∀x(φ ↔ x = y) ↔ ∃z∀x(φ ↔ x = z))

Distinct variable groups:   x,y   x,z   φ,y   φ,z

Allowed substitution hint:   φ(x)

Proof of Theorem eujust

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1596	. . . . 5 ⊢ (y = w → (x = y ↔ x = w))
2	1	bibi2d 221	. . . 4 ⊢ (y = w → ((φ ↔ x = y) ↔ (φ ↔ x = w)))
3	2	albidv 1702	. . 3 ⊢ (y = w → (∀x(φ ↔ x = y) ↔ ∀x(φ ↔ x = w)))
4	3	cbvexv 1792	. 2 ⊢ (∃y∀x(φ ↔ x = y) ↔ ∃w∀x(φ ↔ x = w))
5		equequ2 1596	. . . . 5 ⊢ (w = z → (x = w ↔ x = z))
6	5	bibi2d 221	. . . 4 ⊢ (w = z → ((φ ↔ x = w) ↔ (φ ↔ x = z)))
7	6	albidv 1702	. . 3 ⊢ (w = z → (∀x(φ ↔ x = w) ↔ ∀x(φ ↔ x = z)))
8	7	cbvexv 1792	. 2 ⊢ (∃w∀x(φ ↔ x = w) ↔ ∃z∀x(φ ↔ x = z))
9	4, 8	bitri 173	1 ⊢ (∃y∀x(φ ↔ x = y) ↔ ∃z∀x(φ ↔ x = z))

Syntax hints:   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)