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Theorem pm13.183 2794
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 (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 2122 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
2 eqeq2 2125 . . . 4 (𝑦 = 𝐴 → (𝑧 = 𝑦𝑧 = 𝐴))
32bibi1d 232 . . 3 (𝑦 = 𝐴 → ((𝑧 = 𝑦𝑧 = 𝐵) ↔ (𝑧 = 𝐴𝑧 = 𝐵)))
43albidv 1778 . 2 (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
5 eqeq2 2125 . . . 4 (𝑦 = 𝐵 → (𝑧 = 𝑦𝑧 = 𝐵))
65alrimiv 1828 . . 3 (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦𝑧 = 𝐵))
7 stdpc4 1731 . . . 4 (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵))
8 sbbi 1908 . . . . 5 ([𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9 eqsb3 2219 . . . . . . 7 ([𝑦 / 𝑧]𝑧 = 𝐵𝑦 = 𝐵)
109bibi2i 226 . . . . . 6 (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵))
11 equsb1 1741 . . . . . . 7 [𝑦 / 𝑧]𝑧 = 𝑦
12 bi1 117 . . . . . . 7 (([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵) → ([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵))
1311, 12mpi 15 . . . . . 6 (([𝑦 / 𝑧]𝑧 = 𝑦𝑦 = 𝐵) → 𝑦 = 𝐵)
1410, 13sylbi 120 . . . . 5 (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
158, 14sylbi 120 . . . 4 ([𝑦 / 𝑧](𝑧 = 𝑦𝑧 = 𝐵) → 𝑦 = 𝐵)
167, 15syl 14 . . 3 (∀𝑧(𝑧 = 𝑦𝑧 = 𝐵) → 𝑦 = 𝐵)
176, 16impbii 125 . 2 (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦𝑧 = 𝐵))
181, 4, 17vtoclbg 2719 1 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴𝑧 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312   = wceq 1314  wcel 1463  [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  mpo2eqb  5846
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