Theorem pm13.183 2732

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 2087	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqeq2 2090	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐴))
3	2	bibi1d 231	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
4	3	albidv 1745	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
5		eqeq2 2090	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
6	5	alrimiv 1795	. . 3 ⊢ (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
7		stdpc4 1698	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
8		sbbi 1874	. . . . 5 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9		eqsb3 2182	. . . . . . 7 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ 𝑦 = 𝐵)
10	9	bibi2i 225	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵))
11		equsb1 1708	. . . . . . 7 ⊢ [𝑦 / 𝑧]𝑧 = 𝑦
12		bi1 116	. . . . . . 7 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → ([𝑦 / 𝑧]𝑧 = 𝑦 → 𝑦 = 𝐵))
13	11, 12	mpi 15	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → 𝑦 = 𝐵)
14	10, 13	sylbi 119	. . . . 5 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
15	8, 14	sylbi 119	. . . 4 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
16	7, 15	syl 14	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
17	6, 16	impbii 124	. 2 ⊢ (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
18	1, 4, 17	vtoclbg 2659	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  mpt22eqb  5630