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Theorem mosubt 2712
 Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
Assertion
Ref Expression
mosubt (y∃*xφ∃*xy(y = A φ))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem mosubt
StepHypRef Expression
1 eueq 2706 . . . . . 6 (A V ↔ ∃!y y = A)
2 isset 2555 . . . . . 6 (A V ↔ y y = A)
31, 2bitr3i 175 . . . . 5 (∃!y y = Ay y = A)
4 nfv 1418 . . . . . 6 x y = A
54euexex 1982 . . . . 5 ((∃!y y = A y∃*xφ) → ∃*xy(y = A φ))
63, 5sylanbr 269 . . . 4 ((y y = A y∃*xφ) → ∃*xy(y = A φ))
76expcom 109 . . 3 (y∃*xφ → (y y = A∃*xy(y = A φ)))
8 moanimv 1972 . . 3 (∃*x(y y = A y(y = A φ)) ↔ (y y = A∃*xy(y = A φ)))
97, 8sylibr 137 . 2 (y∃*xφ∃*x(y y = A y(y = A φ)))
10 simpl 102 . . . . 5 ((y = A φ) → y = A)
1110eximi 1488 . . . 4 (y(y = A φ) → y y = A)
1211ancri 307 . . 3 (y(y = A φ) → (y y = A y(y = A φ)))
1312moimi 1962 . 2 (∃*x(y y = A y(y = A φ)) → ∃*xy(y = A φ))
149, 13syl 14 1 (y∃*xφ∃*xy(y = A φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  Vcvv 2551 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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  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-v 2553 This theorem is referenced by:  mosub  2713
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