Theorem sbceqal 3098

Description: Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon, 28-Jun-2011.)

Assertion

Ref	Expression
sbceqal	⊢ (A ∈ V → (∀x(x = A → x = B) → A = B))

Distinct variable groups:   x,B   x,A

Allowed substitution hint:   V(x)

Proof of Theorem sbceqal

Step	Hyp	Ref	Expression
1		spsbc 3059	. 2 ⊢ (A ∈ V → (∀x(x = A → x = B) → [̣A / x]̣(x = A → x = B)))
2		sbcimg 3088	. . 3 ⊢ (A ∈ V → ([̣A / x]̣(x = A → x = B) ↔ ([̣A / x]̣x = A → [̣A / x]̣x = B)))
3		eqid 2353	. . . . 5 ⊢ A = A
4		eqsbc1 3086	. . . . 5 ⊢ (A ∈ V → ([̣A / x]̣x = A ↔ A = A))
5	3, 4	mpbiri 224	. . . 4 ⊢ (A ∈ V → [̣A / x]̣x = A)
6		pm5.5 326	. . . 4 ⊢ ([̣A / x]̣x = A → (([̣A / x]̣x = A → [̣A / x]̣x = B) ↔ [̣A / x]̣x = B))
7	5, 6	syl 15	. . 3 ⊢ (A ∈ V → (([̣A / x]̣x = A → [̣A / x]̣x = B) ↔ [̣A / x]̣x = B))
8		eqsbc1 3086	. . 3 ⊢ (A ∈ V → ([̣A / x]̣x = B ↔ A = B))
9	2, 7, 8	3bitrd 270	. 2 ⊢ (A ∈ V → ([̣A / x]̣(x = A → x = B) ↔ A = B))
10	1, 9	sylibd 205	1 ⊢ (A ∈ V → (∀x(x = A → x = B) → A = B))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

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:  sbeqalb  3099