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Theorem mosubopt 4330
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 1417 . . 3 yyz∃*xφ
2 nfe1 1367 . . . 4 yyz(A = ⟨y, z φ)
32nfmo 1903 . . 3 y∃*xyz(A = ⟨y, z φ)
4 nfa1 1417 . . . . 5 zz∃*xφ
5 nfe1 1367 . . . . . . 7 zz(A = ⟨y, z φ)
65nfex 1512 . . . . . 6 zyz(A = ⟨y, z φ)
76nfmo 1903 . . . . 5 z∃*xyz(A = ⟨y, z φ)
8 copsexg 3954 . . . . . . . 8 (A = ⟨y, z⟩ → (φyz(A = ⟨y, z φ)))
98mobidv 1919 . . . . . . 7 (A = ⟨y, z⟩ → (∃*xφ∃*xyz(A = ⟨y, z φ)))
109biimpcd 148 . . . . . 6 (∃*xφ → (A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
1110sps 1413 . . . . 5 (z∃*xφ → (A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
124, 7, 11exlimd 1472 . . . 4 (z∃*xφ → (z A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
1312sps 1413 . . 3 (yz∃*xφ → (z A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
141, 3, 13exlimd 1472 . 2 (yz∃*xφ → (yz A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
15 moanimv 1958 . . 3 (∃*x(yz A = ⟨y, z yz(A = ⟨y, z φ)) ↔ (yz A = ⟨y, z⟩ → ∃*xyz(A = ⟨y, z φ)))
16 ax-ia1 99 . . . . . 6 ((A = ⟨y, z φ) → A = ⟨y, z⟩)
17162eximi 1476 . . . . 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 1948 . . 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 1226   = wceq 1228  wex 1363  ∃*wmo 1884  cop 3352
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358
This theorem is referenced by:  mosubop  4331  funoprabg  5521
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