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Theorem pm13.183 2979
 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 2359 . 2 (y = A → (y = BA = B))
2 eqeq2 2362 . . . 4 (y = A → (z = yz = A))
32bibi1d 310 . . 3 (y = A → ((z = yz = B) ↔ (z = Az = B)))
43albidv 1625 . 2 (y = A → (z(z = yz = B) ↔ z(z = Az = B)))
5 eqeq2 2362 . . . 4 (y = B → (z = yz = B))
65alrimiv 1631 . . 3 (y = Bz(z = yz = B))
7 stdpc4 2024 . . . 4 (z(z = yz = B) → [y / z](z = yz = B))
8 sbbi 2071 . . . . 5 ([y / z](z = yz = B) ↔ ([y / z]z = y ↔ [y / z]z = B))
9 eqsb3 2454 . . . . . . 7 ([y / z]z = By = B)
109bibi2i 304 . . . . . 6 (([y / z]z = y ↔ [y / z]z = B) ↔ ([y / z]z = yy = B))
11 equsb1 2034 . . . . . . 7 [y / z]z = y
12 bi1 178 . . . . . . 7 (([y / z]z = yy = B) → ([y / z]z = yy = B))
1311, 12mpi 16 . . . . . 6 (([y / z]z = yy = B) → y = B)
1410, 13sylbi 187 . . . . 5 (([y / z]z = y ↔ [y / z]z = B) → y = B)
158, 14sylbi 187 . . . 4 ([y / z](z = yz = B) → y = B)
167, 15syl 15 . . 3 (z(z = yz = B) → y = B)
176, 16impbii 180 . 2 (y = Bz(z = yz = B))
181, 4, 17vtoclbg 2915 1 (A V → (A = Bz(z = Az = B)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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