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Theorem opelopabt 4699
 Description: Closed theorem form of opelopab 4708. (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 4696 . 2 (A, B {x, y φ} ↔ xy(A, B = x, y φ))
2 19.26-2 1594 . . . . 5 (xy((x = A → (φψ)) (y = B → (ψχ))) ↔ (xy(x = A → (φψ)) xy(y = B → (ψχ))))
3 prth 554 . . . . . . 7 (((x = A → (φψ)) (y = B → (ψχ))) → ((x = A y = B) → ((φψ) (ψχ))))
4 bitr 689 . . . . . . 7 (((φψ) (ψχ)) → (φχ))
53, 4syl6 29 . . . . . 6 (((x = A → (φψ)) (y = B → (ψχ))) → ((x = A y = B) → (φχ)))
652alimi 1560 . . . . 5 (xy((x = A → (φψ)) (y = B → (ψχ))) → xy((x = A y = B) → (φχ)))
72, 6sylbir 204 . . . 4 ((xy(x = A → (φψ)) xy(y = B → (ψχ))) → xy((x = A y = B) → (φχ)))
8 copsex2t 4608 . . . 4 ((xy((x = A y = B) → (φχ)) (A V B W)) → (xy(A, B = x, y φ) ↔ χ))
97, 8sylan 457 . . 3 (((xy(x = A → (φψ)) xy(y = B → (ψχ))) (A V B W)) → (xy(A, B = x, y φ) ↔ χ))
1093impa 1146 . 2 ((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A V B W)) → (xy(A, B = x, y φ) ↔ χ))
111, 10syl5bb 248 1 ((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A V B W)) → (A, B {x, y φ} ↔ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623 This theorem is referenced by:  fvopab4t  5385
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