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Theorem mosubt 2834
Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
Assertion
Ref Expression
mosubt (∀𝑦∃*𝑥𝜑 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mosubt
StepHypRef Expression
1 eueq 2828 . . . . . 6 (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴)
2 isset 2666 . . . . . 6 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
31, 2bitr3i 185 . . . . 5 (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
4 nfv 1493 . . . . . 6 𝑥 𝑦 = 𝐴
54euexex 2062 . . . . 5 ((∃!𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
63, 5sylanbr 283 . . . 4 ((∃𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
76expcom 115 . . 3 (∀𝑦∃*𝑥𝜑 → (∃𝑦 𝑦 = 𝐴 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑)))
8 moanimv 2052 . . 3 (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴𝜑)) ↔ (∃𝑦 𝑦 = 𝐴 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑)))
97, 8sylibr 133 . 2 (∀𝑦∃*𝑥𝜑 → ∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴𝜑)))
10 simpl 108 . . . . 5 ((𝑦 = 𝐴𝜑) → 𝑦 = 𝐴)
1110eximi 1564 . . . 4 (∃𝑦(𝑦 = 𝐴𝜑) → ∃𝑦 𝑦 = 𝐴)
1211ancri 322 . . 3 (∃𝑦(𝑦 = 𝐴𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴𝜑)))
1312moimi 2042 . 2 (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴𝜑)) → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
149, 13syl 14 1 (∀𝑦∃*𝑥𝜑 → ∃*𝑥𝑦(𝑦 = 𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314   = wceq 1316  wex 1453  wcel 1465  ∃!weu 1977  ∃*wmo 1978  Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  mosub  2835
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