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Theorem 2reurex 39739
Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2436. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
Assertion
Ref Expression
2reurex (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reurex
StepHypRef Expression
1 reu5 3040 . 2 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑))
2 rexcom 2984 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
3 nfcv 2655 . . . . . 6 𝑦𝐴
4 nfre1 2892 . . . . . 6 𝑦𝑦𝐵 𝜑
53, 4nfrmo 2998 . . . . 5 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
6 rspe 2890 . . . . . . . . . . 11 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
76ex 448 . . . . . . . . . 10 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
87ralrimivw 2854 . . . . . . . . 9 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
9 rmoim 3278 . . . . . . . . 9 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
108, 9syl 17 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
1110impcom 444 . . . . . . 7 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → ∃*𝑥𝐴 𝜑)
12 rmo5 3043 . . . . . . 7 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1311, 12sylib 206 . . . . . 6 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1413ex 448 . . . . 5 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑)))
155, 14reximdai 2899 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑦𝐵𝑥𝐴 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
162, 15syl5bi 230 . . 3 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
1716impcom 444 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑) → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
181, 17sylbi 205 1 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1938  wral 2800  wrex 2801  ∃!wreu 2802  ∃*wrmo 2803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808
This theorem is referenced by:  2rexreu  39743
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