Theorem pm13.183 2902

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 2203	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqeq2 2206	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐴))
3	2	bibi1d 233	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
4	3	albidv 1838	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
5		eqeq2 2206	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
6	5	alrimiv 1888	. . 3 ⊢ (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
7		stdpc4 1789	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
8		sbbi 1978	. . . . 5 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9		eqsb1 2300	. . . . . . 7 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ 𝑦 = 𝐵)
10	9	bibi2i 227	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵))
11		equsb1 1799	. . . . . . 7 ⊢ [𝑦 / 𝑧]𝑧 = 𝑦
12		biimp 118	. . . . . . 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 121	. . . . 5 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
15	8, 14	sylbi 121	. . . 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 126	. 2 ⊢ (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
18	1, 4, 17	vtoclbg 2825	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  [wsb 1776   ∈ wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  mpo2eqb  6032