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Theorem pm13.183 2654
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 2024 . 2 (y = A → (y = BA = B))
2 eqeq2 2027 . . . 4 (y = A → (z = yz = A))
32bibi1d 222 . . 3 (y = A → ((z = yz = B) ↔ (z = Az = B)))
43albidv 1683 . 2 (y = A → (z(z = yz = B) ↔ z(z = Az = B)))
5 eqeq2 2027 . . . 4 (y = B → (z = yz = B))
65alrimiv 1732 . . 3 (y = Bz(z = yz = B))
7 stdpc4 1636 . . . 4 (z(z = yz = B) → [y / z](z = yz = B))
8 sbbi 1811 . . . . 5 ([y / z](z = yz = B) ↔ ([y / z]z = y ↔ [y / z]z = B))
9 eqsb3 2119 . . . . . . 7 ([y / z]z = By = B)
109bibi2i 216 . . . . . 6 (([y / z]z = y ↔ [y / z]z = B) ↔ ([y / z]z = yy = B))
11 equsb1 1646 . . . . . . 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 2587 1 (A 𝑉 → (A = Bz(z = Az = B)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224   = wceq 1226   wcel 1370  [wsb 1623
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533
This theorem is referenced by:  mpt22eqb  5529
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