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Theorem mosubt 2861
 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 2855 . . . . . 6 (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴)
2 isset 2692 . . . . . 6 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
31, 2bitr3i 185 . . . . 5 (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
4 nfv 1508 . . . . . 6 𝑥 𝑦 = 𝐴
54euexex 2084 . . . . 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 2074 . . 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 1579 . . . 4 (∃𝑦(𝑦 = 𝐴𝜑) → ∃𝑦 𝑦 = 𝐴)
1211ancri 322 . . 3 (∃𝑦(𝑦 = 𝐴𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴𝜑)))
1312moimi 2064 . 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 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃!weu 1999  ∃*wmo 2000  Vcvv 2686 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688 This theorem is referenced by:  mosub  2862
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