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Theorem 2mos 2678
Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
Hypothesis
Ref Expression
2mos.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
2mos (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑦,𝜓   𝑥,𝑧,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem 2mos
StepHypRef Expression
1 2mo 2677 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
2 nfv 1980 . . . . . . 7 𝑥𝜓
3 2mos.1 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
43sbiedv 2535 . . . . . . 7 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑𝜓))
52, 4sbie 2533 . . . . . 6 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜓)
65anbi2i 732 . . . . 5 ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑𝜓))
76imbi1i 338 . . . 4 (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
872albii 1885 . . 3 (∀𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
982albii 1885 . 2 (∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
101, 9bitri 264 1 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1618  wex 1841  [wsb 2034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600
This theorem is referenced by: (None)
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