Theorem pm13.183 2980

Description: Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only A is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.183	⊢ (A ∈ V → (A = B ↔ ∀z(z = A ↔ z = B)))

Distinct variable groups:   z,A   z,B

Allowed substitution hint:   V(z)

Proof of Theorem pm13.183

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. 2 ⊢ (y = A → (y = B ↔ A = B))
2		eqeq2 2362	. . . 4 ⊢ (y = A → (z = y ↔ z = A))
3	2	bibi1d 310	. . 3 ⊢ (y = A → ((z = y ↔ z = B) ↔ (z = A ↔ z = B)))
4	3	albidv 1625	. 2 ⊢ (y = A → (∀z(z = y ↔ z = B) ↔ ∀z(z = A ↔ z = B)))
5		eqeq2 2362	. . . 4 ⊢ (y = B → (z = y ↔ z = B))
6	5	alrimiv 1631	. . 3 ⊢ (y = B → ∀z(z = y ↔ z = B))
7		stdpc4 2024	. . . 4 ⊢ (∀z(z = y ↔ z = B) → [y / z](z = y ↔ z = B))
8		sbbi 2071	. . . . 5 ⊢ ([y / z](z = y ↔ z = B) ↔ ([y / z]z = y ↔ [y / z]z = B))
9		eqsb1 2454	. . . . . . 7 ⊢ ([y / z]z = B ↔ y = B)
10	9	bibi2i 304	. . . . . 6 ⊢ (([y / z]z = y ↔ [y / z]z = B) ↔ ([y / z]z = y ↔ y = B))
11		equsb1 2034	. . . . . . 7 ⊢ [y / z]z = y
12		bi1 178	. . . . . . 7 ⊢ (([y / z]z = y ↔ y = B) → ([y / z]z = y → y = B))
13	11, 12	mpi 16	. . . . . 6 ⊢ (([y / z]z = y ↔ y = B) → y = B)
14	10, 13	sylbi 187	. . . . 5 ⊢ (([y / z]z = y ↔ [y / z]z = B) → y = B)
15	8, 14	sylbi 187	. . . 4 ⊢ ([y / z](z = y ↔ z = B) → y = B)
16	7, 15	syl 15	. . 3 ⊢ (∀z(z = y ↔ z = B) → y = B)
17	6, 16	impbii 180	. 2 ⊢ (y = B ↔ ∀z(z = y ↔ z = B))
18	1, 4, 17	vtoclbg 2916	1 ⊢ (A ∈ V → (A = B ↔ ∀z(z = A ↔ z = B)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710

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  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by: (None)