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Theorem mosubopt 4348
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
mosubopt (yz∃*xφ∃*xyz(A = ⟨y, z φ))
Distinct variable group:   x,y,z,A
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem mosubopt
StepHypRef Expression
1 nfa1 1431 . . 3 yyz∃*xφ
2 nfe1 1382 . . . 4 yyz(A = ⟨y, z φ)
32nfmo 1917 . . 3 y∃*xyz(A = ⟨y, z φ)
4 nfa1 1431 . . . . 5 zz∃*xφ
5 nfe1 1382 . . . . . . 7 zz(A = ⟨y, z φ)
65nfex 1525 . . . . . 6 zyz(A = ⟨y, z φ)
76nfmo 1917 . . . . 5 z∃*xyz(A = ⟨y, z φ)
8 copsexg 3972 . . . . . . . 8 (A = ⟨y, z⟩ → (φyz(A = ⟨y, z φ)))
98mobidv 1933 . . . . . . 7 (A = ⟨y, z⟩ → (∃*xφ∃*xyz(A = ⟨y, z φ)))
109biimpcd 148 . . . . . 6 (∃*xφ → (A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
1110sps 1427 . . . . 5 (z∃*xφ → (A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
124, 7, 11exlimd 1485 . . . 4 (z∃*xφ → (z A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
1312sps 1427 . . 3 (yz∃*xφ → (z A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
141, 3, 13exlimd 1485 . 2 (yz∃*xφ → (yz A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
15 moanimv 1972 . . 3 (∃*x(yz A = ⟨y, z yz(A = ⟨y, z φ)) ↔ (yz A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
16 simpl 102 . . . . . 6 ((A = ⟨y, z φ) → A = ⟨y, z⟩)
17162eximi 1489 . . . . 5 (yz(A = ⟨y, z φ) → yz A = ⟨y, z⟩)
1817ancri 307 . . . 4 (yz(A = ⟨y, z φ) → (yz A = ⟨y, z yz(A = ⟨y, z φ)))
1918moimi 1962 . . 3 (∃*x(yz A = ⟨y, z yz(A = ⟨y, z φ)) → ∃*xyz(A = ⟨y, z φ))
2015, 19sylbir 125 . 2 ((yz A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)) → ∃*xyz(A = ⟨y, z φ))
2114, 20syl 14 1 (yz∃*xφ∃*xyz(A = ⟨y, z φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378  ∃*wmo 1898  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:  mosubop  4349  funoprabg  5542
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