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Theorem 2mos 2733
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 2732 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
2 nfv 1915 . . . . . . . . . 10 𝑦 𝑥 = 𝑧
32sbrim 2313 . . . . . . . . 9 ([𝑤 / 𝑦](𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
4 2mos.1 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
54expcom 416 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
65pm5.74d 275 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
76sbievw 2103 . . . . . . . . 9 ([𝑤 / 𝑦](𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓))
83, 7bitr3i 279 . . . . . . . 8 ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧𝜓))
98pm5.74ri 274 . . . . . . 7 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑𝜓))
109sbievw 2103 . . . . . 6 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜓)
1110anbi2i 624 . . . . 5 ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑𝜓))
1211imbi1i 352 . . . 4 (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
13122albii 1821 . . 3 (∀𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
14132albii 1821 . 2 (∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
151, 14bitri 277 1 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1535  wex 1780  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622
This theorem is referenced by: (None)
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