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Theorem 2eu7 2695
Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
Assertion
Ref Expression
2eu7 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Proof of Theorem 2eu7
StepHypRef Expression
1 nfe1 2174 . . . 4 𝑥𝑥𝜑
21nfeu 2621 . . 3 𝑥∃!𝑦𝑥𝜑
32euan 2666 . 2 (∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
4 ancom 465 . . . . 5 ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
54eubii 2627 . . . 4 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6 nfe1 2174 . . . . 5 𝑦𝑦𝜑
76euan 2666 . . . 4 (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
8 ancom 465 . . . 4 ((∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
95, 7, 83bitri 286 . . 3 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
109eubii 2627 . 2 (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
11 ancom 465 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
123, 10, 113bitr4ri 293 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1851  ∃!weu 2605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-eu 2609
This theorem is referenced by:  2eu8  2696
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