Theorem pm13.183 3597

Description: Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) Avoid ax-13 2372. (Revised by Wolf Lammen, 29-Apr-2023.)

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 2742	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqeq2 2750	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐴))
3	2	bibi1d 344	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
4	3	albidv 1923	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
5		eqeq2 2750	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
6	5	alrimiv 1930	. . 3 ⊢ (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
7		stdpc4 2071	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
8		sbbi 2305	. . . . 5 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9		eqsb1 2865	. . . . . . 7 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ 𝑦 = 𝐵)
10	9	bibi2i 338	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵))
11		equsb1v 2103	. . . . . . 7 ⊢ [𝑦 / 𝑧]𝑧 = 𝑦
12		biimp 214	. . . . . . 7 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → ([𝑦 / 𝑧]𝑧 = 𝑦 → 𝑦 = 𝐵))
13	11, 12	mpi 20	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → 𝑦 = 𝐵)
14	10, 13	sylbi 216	. . . . 5 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
15	8, 14	sylbi 216	. . . 4 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
16	7, 15	syl 17	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
17	6, 16	impbii 208	. 2 ⊢ (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
18	1, 4, 17	vtoclbg 3507	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  [wsb 2067   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816

This theorem is referenced by:  mpo2eqb  7406