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Theorem copsex2g 3974
 Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
Hypothesis
Ref Expression
copsex2g.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
copsex2g ((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 copsex2g
StepHypRef Expression
1 elisset 2562 . 2 (A 𝑉x x = A)
2 elisset 2562 . 2 (B 𝑊y y = B)
3 eeanv 1804 . . 3 (xy(x = A y = B) ↔ (x x = A y y = B))
4 nfe1 1382 . . . . 5 xxy(⟨A, B⟩ = ⟨x, y φ)
5 nfv 1418 . . . . 5 xψ
64, 5nfbi 1478 . . . 4 x(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
7 nfe1 1382 . . . . . . 7 yy(⟨A, B⟩ = ⟨x, y φ)
87nfex 1525 . . . . . 6 yxy(⟨A, B⟩ = ⟨x, y φ)
9 nfv 1418 . . . . . 6 yψ
108, 9nfbi 1478 . . . . 5 y(xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ)
11 opeq12 3542 . . . . . . 7 ((x = A y = B) → ⟨x, y⟩ = ⟨A, B⟩)
12 copsexg 3972 . . . . . . . 8 (⟨A, B⟩ = ⟨x, y⟩ → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1312eqcoms 2040 . . . . . . 7 (⟨x, y⟩ = ⟨A, B⟩ → (φxy(⟨A, B⟩ = ⟨x, y φ)))
1411, 13syl 14 . . . . . 6 ((x = A y = B) → (φxy(⟨A, B⟩ = ⟨x, y φ)))
15 copsex2g.1 . . . . . 6 ((x = A y = B) → (φψ))
1614, 15bitr3d 179 . . . . 5 ((x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
1710, 16exlimi 1482 . . . 4 (y(x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
186, 17exlimi 1482 . . 3 (xy(x = A y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
193, 18sylbir 125 . 2 ((x x = A y y = B) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
201, 2, 19syl2an 273 1 ((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   = 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:  opelopabga  3991  ov6g  5580  ltresr  6716
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