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Theorem 2eu7 2720
 Description: Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 19-Feb-2005.) (New usage is discouraged.)
Assertion
Ref Expression
2eu7 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Proof of Theorem 2eu7
StepHypRef Expression
1 nfe1 2151 . . . 4 𝑥𝑥𝜑
21nfeu 2655 . . 3 𝑥∃!𝑦𝑥𝜑
32euan 2683 . 2 (∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
4 ancom 464 . . . . 5 ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
54eubii 2645 . . . 4 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6 nfe1 2151 . . . . 5 𝑦𝑦𝜑
76euan 2683 . . . 4 (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
8 ancom 464 . . . 4 ((∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
95, 7, 83bitri 300 . . 3 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
109eubii 2645 . 2 (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
11 ancom 464 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
123, 10, 113bitr4ri 307 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781  ∃!weu 2628 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-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629 This theorem is referenced by:  2eu8  2721
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