New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ceqsralt GIF version

Theorem ceqsralt 2882
 Description: Restricted quantifier version of ceqsalt 2881. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsralt ((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem ceqsralt
StepHypRef Expression
1 df-ral 2619 . . . 4 (x B (x = Aφ) ↔ x(x B → (x = Aφ)))
2 eleq1 2413 . . . . . . . . 9 (x = A → (x BA B))
32pm5.32ri 619 . . . . . . . 8 ((x B x = A) ↔ (A B x = A))
43imbi1i 315 . . . . . . 7 (((x B x = A) → φ) ↔ ((A B x = A) → φ))
5 impexp 433 . . . . . . 7 (((x B x = A) → φ) ↔ (x B → (x = Aφ)))
6 impexp 433 . . . . . . 7 (((A B x = A) → φ) ↔ (A B → (x = Aφ)))
74, 5, 63bitr3i 266 . . . . . 6 ((x B → (x = Aφ)) ↔ (A B → (x = Aφ)))
87albii 1566 . . . . 5 (x(x B → (x = Aφ)) ↔ x(A B → (x = Aφ)))
98a1i 10 . . . 4 ((Ⅎxψ x(x = A → (φψ)) A B) → (x(x B → (x = Aφ)) ↔ x(A B → (x = Aφ))))
101, 9syl5bb 248 . . 3 ((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ x(A B → (x = Aφ))))
11 19.21v 1890 . . 3 (x(A B → (x = Aφ)) ↔ (A Bx(x = Aφ)))
1210, 11syl6bb 252 . 2 ((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ (A Bx(x = Aφ))))
13 biimt 325 . . 3 (A B → (x(x = Aφ) ↔ (A Bx(x = Aφ))))
14133ad2ant3 978 . 2 ((Ⅎxψ x(x = A → (φψ)) A B) → (x(x = Aφ) ↔ (A Bx(x = Aφ))))
15 ceqsalt 2881 . 2 ((Ⅎxψ x(x = A → (φψ)) A B) → (x(x = Aφ) ↔ ψ))
1612, 14, 153bitr2d 272 1 ((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614 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-ral 2619  df-v 2861 This theorem is referenced by:  ceqsralv  2886
 Copyright terms: Public domain W3C validator