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Theorem 2eu7 2738
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 2145 . . . 4 𝑥𝑥𝜑
21nfeu 2673 . . 3 𝑥∃!𝑦𝑥𝜑
32euan 2699 . 2 (∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
4 ancom 461 . . . . 5 ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
54eubii 2663 . . . 4 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6 nfe1 2145 . . . . 5 𝑦𝑦𝜑
76euan 2699 . . . 4 (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
8 ancom 461 . . . 4 ((∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
95, 7, 83bitri 298 . . 3 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
109eubii 2663 . 2 (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
11 ancom 461 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
123, 10, 113bitr4ri 305 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771  ∃!weu 2646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-mo 2615  df-eu 2647
This theorem is referenced by:  2eu8  2739
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