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Theorem 2reurex 3757
Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2638. (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 3379 . 2 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑))
2 rexcom 3288 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
3 nfcv 2904 . . . . . 6 𝑦𝐴
4 nfre1 3283 . . . . . 6 𝑦𝑦𝐵 𝜑
53, 4nfrmow 3410 . . . . 5 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
6 rspe 3247 . . . . . . . . . . 11 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
76ex 414 . . . . . . . . . 10 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
87ralrimivw 3151 . . . . . . . . 9 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
9 rmoim 3737 . . . . . . . . 9 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
108, 9syl 17 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
1110impcom 409 . . . . . . 7 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → ∃*𝑥𝐴 𝜑)
12 rmo5 3397 . . . . . . 7 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1311, 12sylib 217 . . . . . 6 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1413ex 414 . . . . 5 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑)))
155, 14reximdai 3259 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑦𝐵𝑥𝐴 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
162, 15biimtrid 241 . . 3 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
1716impcom 409 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑) → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
181, 17sylbi 216 1 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  ∃*wrmo 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-mo 2535  df-eu 2564  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378
This theorem is referenced by:  2rexreu  3759
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