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Theorem opelopabt 3990
 Description: Closed theorem form of opelopab 3999. (Contributed by NM, 19-Feb-2013.)
Assertion
Ref Expression
opelopabt ((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A 𝑉 B 𝑊)) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
Distinct variable groups:   x,y,A   x,B,y   χ,x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem opelopabt
StepHypRef Expression
1 elopab 3986 . 2 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ xy(⟨A, B⟩ = ⟨x, y φ))
2 19.26-2 1368 . . . . 5 (xy((x = A → (φψ)) (y = B → (ψχ))) ↔ (xy(x = A → (φψ)) xy(y = B → (ψχ))))
3 prth 326 . . . . . . 7 (((x = A → (φψ)) (y = B → (ψχ))) → ((x = A y = B) → ((φψ) (ψχ))))
4 bitr 441 . . . . . . 7 (((φψ) (ψχ)) → (φχ))
53, 4syl6 29 . . . . . 6 (((x = A → (φψ)) (y = B → (ψχ))) → ((x = A y = B) → (φχ)))
652alimi 1342 . . . . 5 (xy((x = A → (φψ)) (y = B → (ψχ))) → xy((x = A y = B) → (φχ)))
72, 6sylbir 125 . . . 4 ((xy(x = A → (φψ)) xy(y = B → (ψχ))) → xy((x = A y = B) → (φχ)))
8 copsex2t 3973 . . . 4 ((xy((x = A y = B) → (φχ)) (A 𝑉 B 𝑊)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ χ))
97, 8sylan 267 . . 3 (((xy(x = A → (φψ)) xy(y = B → (ψχ))) (A 𝑉 B 𝑊)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ χ))
1093impa 1098 . 2 ((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A 𝑉 B 𝑊)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ χ))
111, 10syl5bb 181 1 ((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A 𝑉 B 𝑊)) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370  {copab 3808 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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810 This theorem is referenced by: (None)
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