Theorem sbeqalb 3099

Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Assertion

Ref	Expression
sbeqalb	⊢ (A ∈ V → ((∀x(φ ↔ x = A) ∧ ∀x(φ ↔ x = B)) → A = B))

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem sbeqalb

Step	Hyp	Ref	Expression
1		bibi1 317	. . . . 5 ⊢ ((φ ↔ x = A) → ((φ ↔ x = B) ↔ (x = A ↔ x = B)))
2	1	biimpa 470	. . . 4 ⊢ (((φ ↔ x = A) ∧ (φ ↔ x = B)) → (x = A ↔ x = B))
3	2	biimpd 198	. . 3 ⊢ (((φ ↔ x = A) ∧ (φ ↔ x = B)) → (x = A → x = B))
4	3	alanimi 1562	. 2 ⊢ ((∀x(φ ↔ x = A) ∧ ∀x(φ ↔ x = B)) → ∀x(x = A → x = B))
5		sbceqal 3098	. 2 ⊢ (A ∈ V → (∀x(x = A → x = B) → A = B))
6	4, 5	syl5 28	1 ⊢ (A ∈ V → ((∀x(φ ↔ x = A) ∧ ∀x(φ ↔ x = B)) → A = B))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710

This proof depends on 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 proof 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  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is used by:  iotaval  4351