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Theorem 2mos 2652
Description: Double "there 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 2651 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
2 nfv 1920 . . . . . . . . . 10 𝑦 𝑥 = 𝑧
32sbrim 2304 . . . . . . . . 9 ([𝑤 / 𝑦](𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
4 2mos.1 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
54expcom 413 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
65pm5.74d 272 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
76sbievw 2098 . . . . . . . . 9 ([𝑤 / 𝑦](𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓))
83, 7bitr3i 276 . . . . . . . 8 ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧𝜓))
98pm5.74ri 271 . . . . . . 7 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑𝜓))
109sbievw 2098 . . . . . 6 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜓)
1110anbi2i 622 . . . . 5 ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑𝜓))
1211imbi1i 349 . . . 4 (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
13122albii 1826 . . 3 (∀𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
14132albii 1826 . 2 (∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
151, 14bitri 274 1 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1539  wex 1785  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-11 2157  ax-12 2174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541
This theorem is referenced by: (None)
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