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Theorem 2reurex 41604
Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2646. (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 3262 . 2 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑))
2 rexcom 3201 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
3 nfcv 2866 . . . . . 6 𝑦𝐴
4 nfre1 3107 . . . . . 6 𝑦𝑦𝐵 𝜑
53, 4nfrmo 3217 . . . . 5 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
6 rspe 3105 . . . . . . . . . . 11 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
76ex 449 . . . . . . . . . 10 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
87ralrimivw 3069 . . . . . . . . 9 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
9 rmoim 3513 . . . . . . . . 9 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
108, 9syl 17 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
1110impcom 445 . . . . . . 7 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → ∃*𝑥𝐴 𝜑)
12 rmo5 3265 . . . . . . 7 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1311, 12sylib 208 . . . . . 6 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1413ex 449 . . . . 5 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑)))
155, 14reximdai 3114 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑦𝐵𝑥𝐴 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
162, 15syl5bi 232 . . 3 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
1716impcom 445 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑) → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
181, 17sylbi 207 1 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2103  wral 3014  wrex 3015  ∃!wreu 3016  ∃*wrmo 3017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022
This theorem is referenced by:  2rexreu  41608
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