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Theorem mosubt 2694
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 2688 . . . . . 6 (A V ↔ ∃!y y = A)
2 isset 2538 . . . . . 6 (A V ↔ y y = A)
31, 2bitr3i 175 . . . . 5 (∃!y y = Ay y = A)
4 nfv 1404 . . . . . 6 x y = A
54euexex 1968 . . . . 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 1958 . . 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 ax-ia1 99 . . . . 5 ((y = A φ) → y = A)
1110eximi 1475 . . . 4 (y(y = A φ) → y y = A)
1211ancri 307 . . 3 (y(y = A φ) → (y y = A y(y = A φ)))
1312moimi 1947 . 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 1315  wex 1362   = wceq 1374   wcel 1376  ∃!weu 1882  ∃*wmo 1883  Vcvv 2534
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 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1232  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-v 2536
This theorem is referenced by:  mosub  2695
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