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Theorem 2reu8 43607
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2745. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 43606. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
2reu8 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2reu8
StepHypRef Expression
1 2reu2 3854 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵𝑥𝐴 𝜑))
21pm5.32i 578 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
3 nfcv 2979 . . . . 5 𝑥𝐵
4 nfreu1 3351 . . . . 5 𝑥∃!𝑥𝐴 𝜑
53, 4nfreuw 3355 . . . 4 𝑥∃!𝑦𝐵 ∃!𝑥𝐴 𝜑
65reuan 3852 . . 3 (∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
7 ancom 464 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
87reubii 3372 . . . . 5 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
9 nfre1 3292 . . . . . 6 𝑦𝑦𝐵 𝜑
109reuan 3852 . . . . 5 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑))
11 ancom 464 . . . . 5 ((∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
128, 10, 113bitri 300 . . . 4 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1312reubii 3372 . . 3 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
14 ancom 464 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
156, 13, 143bitr4ri 307 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
16 2reu7 43606 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
172, 15, 163bitr3ri 305 1 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wrex 3131  ∃!wreu 3132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138
This theorem is referenced by: (None)
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