ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  copsex2t Structured version   GIF version

Theorem copsex2t 3952
Description: Closed theorem form of copsex2g 3953. (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 2541 . . . 4 (A 𝑉x x = A)
2 elisset 2541 . . . 4 (B 𝑊y y = B)
31, 2anim12i 321 . . 3 ((A 𝑉 B 𝑊) → (x x = A y y = B))
4 eeanv 1785 . . 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 1412 . . . 4 xxy((x = A y = B) → (φψ))
7 nfe1 1362 . . . . 5 xxy(⟨A, B⟩ = ⟨x, y φ)
8 nfv 1398 . . . . 5 xψ
97, 8nfbi 1459 . . . 4 x(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
10 nfa2 1449 . . . . 5 yxy((x = A y = B) → (φψ))
11 nfe1 1362 . . . . . . 7 yy(⟨A, B⟩ = ⟨x, y φ)
1211nfex 1506 . . . . . 6 yxy(⟨A, B⟩ = ⟨x, y φ)
13 nfv 1398 . . . . . 6 yψ
1412, 13nfbi 1459 . . . . 5 y(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
15 opeq12 3521 . . . . . . . . 9 ((x = A y = B) → ⟨x, y⟩ = ⟨A, B⟩)
16 copsexg 3951 . . . . . . . . . 10 (⟨A, B⟩ = ⟨x, y⟩ → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1716eqcoms 2021 . . . . . . . . 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 1378 . . . . . . . . 9 (xy((x = A y = B) → (φψ)) → y((x = A y = B) → (φψ)))
212019.21bi 1428 . . . . . . . 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 1466 . . . 4 (xy((x = A y = B) → (φψ)) → (y(x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)))
266, 9, 25exlimd 1466 . . 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 1224   = wceq 1226  wex 1358   wcel 1370  cop 3349
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355
This theorem is referenced by:  opelopabt  3969
  Copyright terms: Public domain W3C validator