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Theorem opelopabt 3973
Description: Closed theorem form of opelopab 3982. (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 3969 . 2 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ xy(⟨A, B⟩ = ⟨x, y φ))
2 19.26-2 1351 . . . . 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 444 . . . . . . 7 (((φψ) (ψχ)) → (φχ))
53, 4syl6 29 . . . . . 6 (((x = A → (φψ)) (y = B → (ψχ))) → ((x = A y = B) → (φχ)))
652alimi 1325 . . . . 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 3956 . . . 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 1085 . 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 873  wal 1226   = wceq 1228  wex 1362   wcel 1374  cop 3353  {copab 3791
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793
This theorem is referenced by: (None)
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