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Theorem ceqsalt 2881
 Description: Closed theorem version of ceqsalg 2883. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsalt ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) ↔ ψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem ceqsalt
StepHypRef Expression
1 elisset 2869 . . . 4 (A Vx x = A)
213ad2ant3 978 . . 3 ((Ⅎxψ x(x = A → (φψ)) A V) → x x = A)
3 bi1 178 . . . . . . 7 ((φψ) → (φψ))
43imim3i 55 . . . . . 6 ((x = A → (φψ)) → ((x = Aφ) → (x = Aψ)))
54al2imi 1561 . . . . 5 (x(x = A → (φψ)) → (x(x = Aφ) → x(x = Aψ)))
653ad2ant2 977 . . . 4 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) → x(x = Aψ)))
7 19.23t 1800 . . . . 5 (Ⅎxψ → (x(x = Aψ) ↔ (x x = Aψ)))
873ad2ant1 976 . . . 4 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aψ) ↔ (x x = Aψ)))
96, 8sylibd 205 . . 3 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) → (x x = Aψ)))
102, 9mpid 37 . 2 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) → ψ))
11 bi2 189 . . . . . . 7 ((φψ) → (ψφ))
1211imim2i 13 . . . . . 6 ((x = A → (φψ)) → (x = A → (ψφ)))
1312com23 72 . . . . 5 ((x = A → (φψ)) → (ψ → (x = Aφ)))
1413alimi 1559 . . . 4 (x(x = A → (φψ)) → x(ψ → (x = Aφ)))
15143ad2ant2 977 . . 3 ((Ⅎxψ x(x = A → (φψ)) A V) → x(ψ → (x = Aφ)))
16 19.21t 1795 . . . 4 (Ⅎxψ → (x(ψ → (x = Aφ)) ↔ (ψx(x = Aφ))))
17163ad2ant1 976 . . 3 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(ψ → (x = Aφ)) ↔ (ψx(x = Aφ))))
1815, 17mpbid 201 . 2 ((Ⅎxψ x(x = A → (φψ)) A V) → (ψx(x = Aφ)))
1910, 18impbid 183 1 ((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  ceqsralt  2882
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