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Theorem pm13.183 2675
 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 = Bz(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 2043 . 2 (y = A → (y = BA = B))
2 eqeq2 2046 . . . 4 (y = A → (z = yz = A))
32bibi1d 222 . . 3 (y = A → ((z = yz = B) ↔ (z = Az = B)))
43albidv 1702 . 2 (y = A → (z(z = yz = B) ↔ z(z = Az = B)))
5 eqeq2 2046 . . . 4 (y = B → (z = yz = B))
65alrimiv 1751 . . 3 (y = Bz(z = yz = B))
7 stdpc4 1655 . . . 4 (z(z = yz = B) → [y / z](z = yz = B))
8 sbbi 1830 . . . . 5 ([y / z](z = yz = B) ↔ ([y / z]z = y ↔ [y / z]z = B))
9 eqsb3 2138 . . . . . . 7 ([y / z]z = By = B)
109bibi2i 216 . . . . . 6 (([y / z]z = y ↔ [y / z]z = B) ↔ ([y / z]z = yy = B))
11 equsb1 1665 . . . . . . 7 [y / z]z = y
12 bi1 111 . . . . . . 7 (([y / z]z = yy = B) → ([y / z]z = yy = B))
1311, 12mpi 15 . . . . . 6 (([y / z]z = yy = B) → y = B)
1410, 13sylbi 114 . . . . 5 (([y / z]z = y ↔ [y / z]z = B) → y = B)
158, 14sylbi 114 . . . 4 ([y / z](z = yz = B) → y = B)
167, 15syl 14 . . 3 (z(z = yz = B) → y = B)
176, 16impbii 117 . 2 (y = Bz(z = yz = B))
181, 4, 17vtoclbg 2608 1 (A 𝑉 → (A = Bz(z = Az = B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by:  mpt22eqb  5552
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