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Theorem pm13.183 3644
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 2392. (Revised by Wolf Lammen, 29-Apr-2023.)
Assertion
Ref Expression
pm13.183 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 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.
StepHypRef Expression
1 eqeq1 2828 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
2 eqeq2 2836 . . . 4 (𝑦 = 𝐴 → (𝑧 = 𝑦𝑧 = 𝐴))
32bibi1d 347 . . 3 (𝑦 = 𝐴 → ((𝑧 = 𝑦𝑧 = 𝐵) ↔ (𝑧 = 𝐴𝑧 = 𝐵)))
43albidv 1922 . 2 (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
5 eqeq2 2836 . . . 4 (𝑦 = 𝐵 → (𝑧 = 𝑦𝑧 = 𝐵))
65alrimiv 1929 . . 3 (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦𝑧 = 𝐵))
7 stdpc4 2074 . . . 4 (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵))
8 sbbi 2319 . . . . 5 ([𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9 eqsb3 2942 . . . . . . 7 ([𝑦 / 𝑧]𝑧 = 𝐵𝑦 = 𝐵)
109bibi2i 341 . . . . . 6 (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵))
11 equsb1v 2113 . . . . . . 7 [𝑦 / 𝑧]𝑧 = 𝑦
12 biimp 218 . . . . . . 7 (([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵) → ([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵))
1311, 12mpi 20 . . . . . 6 (([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵) → 𝑦 = 𝐵)
1410, 13sylbi 220 . . . . 5 (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
158, 14sylbi 220 . . . 4 ([𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵) → 𝑦 = 𝐵)
167, 15syl 17 . . 3 (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) → 𝑦 = 𝐵)
176, 16impbii 212 . 2 (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦𝑧 = 𝐵))
181, 4, 17vtoclbg 3555 1 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  [wsb 2070  wcel 2115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-cleq 2817  df-clel 2896
This theorem is referenced by:  mpo2eqb  7278
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