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Theorem copsex2t 3973
Description: Closed theorem form of copsex2g 3974. (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 2562 . . . 4 (A 𝑉x x = A)
2 elisset 2562 . . . 4 (B 𝑊y y = B)
31, 2anim12i 321 . . 3 ((A 𝑉 B 𝑊) → (x x = A y y = B))
4 eeanv 1804 . . 3 (xy(x = A y = B) ↔ (x x = A y y = B))
53, 4sylibr 137 . 2 ((A 𝑉 B 𝑊) → xy(x = A y = B))
6 nfa1 1431 . . . 4 xxy((x = A y = B) → (φψ))
7 nfe1 1382 . . . . 5 xxy(⟨A, B⟩ = ⟨x, y φ)
8 nfv 1418 . . . . 5 xψ
97, 8nfbi 1478 . . . 4 x(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
10 nfa2 1468 . . . . 5 yxy((x = A y = B) → (φψ))
11 nfe1 1382 . . . . . . 7 yy(⟨A, B⟩ = ⟨x, y φ)
1211nfex 1525 . . . . . 6 yxy(⟨A, B⟩ = ⟨x, y φ)
13 nfv 1418 . . . . . 6 yψ
1412, 13nfbi 1478 . . . . 5 y(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
15 opeq12 3542 . . . . . . . . 9 ((x = A y = B) → ⟨x, y⟩ = ⟨A, B⟩)
16 copsexg 3972 . . . . . . . . . 10 (⟨A, B⟩ = ⟨x, y⟩ → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1716eqcoms 2040 . . . . . . . . 9 (⟨x, y⟩ = ⟨A, B⟩ → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1815, 17syl 14 . . . . . . . 8 ((x = A y = B) → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1918adantl 262 . . . . . . 7 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (φxy(⟨A, B⟩ = ⟨x, y φ)))
20 sp 1398 . . . . . . . . 9 (xy((x = A y = B) → (φψ)) → y((x = A y = B) → (φψ)))
212019.21bi 1447 . . . . . . . 8 (xy((x = A y = B) → (φψ)) → ((x = A y = B) → (φψ)))
2221imp 115 . . . . . . 7 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (φψ))
2319, 22bitr3d 179 . . . . . 6 ((xy((x = A y = B) → (φψ)) (x = A y = B)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
2423ex 108 . . . . 5 (xy((x = A y = B) → (φψ)) → ((x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)))
2510, 14, 24exlimd 1485 . . . 4 (xy((x = A y = B) → (φψ)) → (y(x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)))
266, 9, 25exlimd 1485 . . 3 (xy((x = A y = B) → (φψ)) → (xy(x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)))
2726imp 115 . 2 ((xy((x = A y = B) → (φψ)) xy(x = A y = B)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
285, 27sylan2 270 1 ((xy((x = A y = B) → (φψ)) (A 𝑉 B 𝑊)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  cop 3370
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-bnd 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
This theorem is referenced by:  opelopabt  3990
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