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Theorem copsex2t 4608
 Description: Closed theorem form of copsex2g 4609. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
copsex2t ((xy((x = A y = B) → (φψ)) (A V B W)) → (xy(A, B = x, y φ) ↔ ψ))
Distinct variable groups:   x,y,ψ   x,A,y   x,B,y
Allowed substitution hints:   φ(x,y)   V(x,y)   W(x,y)

Proof of Theorem copsex2t
StepHypRef Expression
1 elisset 2869 . . . 4 (A Vx x = A)
2 elisset 2869 . . . 4 (B Wy y = B)
31, 2anim12i 549 . . 3 ((A V B W) → (x x = A y y = B))
4 eeanv 1913 . . 3 (xy(x = A y = B) ↔ (x x = A y y = B))
53, 4sylibr 203 . 2 ((A V B W) → xy(x = A y = B))
6 nfa1 1788 . . . 4 xxy((x = A y = B) → (φψ))
7 nfe1 1732 . . . . 5 xxy(A, B = x, y φ)
8 nfv 1619 . . . . 5 xψ
97, 8nfbi 1834 . . . 4 x(xy(A, B = x, y φ) ↔ ψ)
10 nfa2 1855 . . . . 5 yxy((x = A y = B) → (φψ))
11 nfe1 1732 . . . . . . 7 yy(A, B = x, y φ)
1211nfex 1843 . . . . . 6 yxy(A, B = x, y φ)
13 nfv 1619 . . . . . 6 yψ
1412, 13nfbi 1834 . . . . 5 y(xy(A, B = x, y φ) ↔ ψ)
15 opeq12 4580 . . . . . . . . 9 ((x = A y = B) → x, y = A, B)
16 copsexg 4607 . . . . . . . . . 10 (A, B = x, y → (φxy(A, B = x, y φ)))
1716eqcoms 2356 . . . . . . . . 9 (x, y = A, B → (φxy(A, B = x, y φ)))
1815, 17syl 15 . . . . . . . 8 ((x = A y = B) → (φxy(A, B = x, y φ)))
1918adantl 452 . . . . . . 7 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (φxy(A, B = x, y φ)))
20 sp 1747 . . . . . . . . 9 (xy((x = A y = B) → (φψ)) → y((x = A y = B) → (φψ)))
212019.21bi 1758 . . . . . . . 8 (xy((x = A y = B) → (φψ)) → ((x = A y = B) → (φψ)))
2221imp 418 . . . . . . 7 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (φψ))
2319, 22bitr3d 246 . . . . . 6 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (xy(A, B = x, y φ) ↔ ψ))
2423ex 423 . . . . 5 (xy((x = A y = B) → (φψ)) → ((x = A y = B) → (xy(A, B = x, y φ) ↔ ψ)))
2510, 14, 24exlimd 1806 . . . 4 (xy((x = A y = B) → (φψ)) → (y(x = A y = B) → (xy(A, B = x, y φ) ↔ ψ)))
266, 9, 25exlimd 1806 . . 3 (xy((x = A y = B) → (φψ)) → (xy(x = A y = B) → (xy(A, B = x, y φ) ↔ ψ)))
2726imp 418 . 2 ((xy((x = A y = B) → (φψ)) xy(x = A y = B)) → (xy(A, B = x, y φ) ↔ ψ))
285, 27sylan2 460 1 ((xy((x = A y = B) → (φψ)) (A V B W)) → (xy(A, B = x, y φ) ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561 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 This theorem is referenced by:  opelopabt  4699
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