Theorem pm13.183 3666

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 2375. (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 2739	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqeq2 2747	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐴))
3	2	bibi1d 343	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
4	3	albidv 1918	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
5		eqeq2 2747	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
6	5	alrimiv 1925	. . 3 ⊢ (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
7		stdpc4 2066	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
8		sbbi 2307	. . . . 5 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9		equsb1v 2103	. . . . . . 7 ⊢ [𝑦 / 𝑧]𝑧 = 𝑦
10	9	tbt 369	. . . . . 6 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ [𝑦 / 𝑧]𝑧 = 𝑦))
11		bicom 222	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝐵 ↔ [𝑦 / 𝑧]𝑧 = 𝑦) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
12	10, 11	bitri 275	. . . . 5 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
13		eqsb1 2865	. . . . 5 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ 𝑦 = 𝐵)
14	8, 12, 13	3bitr2i 299	. . . 4 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ 𝑦 = 𝐵)
15	7, 14	sylib 218	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
16	6, 15	impbii 209	. 2 ⊢ (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
17	1, 4, 16	vtoclbg 3557	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537  [wsb 2062   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814

This theorem is referenced by:  mpo2eqb  7565