Theorem eupickb 2639

Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Assertion

Ref	Expression
eupickb	⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))

Proof of Theorem eupickb

Step	Hyp	Ref	Expression
1		eupick 2637	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
2	1	3adant2 1137	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3		exancom 1868	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4		eupick 2637	. . . 4 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓 ∧ 𝜑)) → (𝜓 → 𝜑))
5	3, 4	sylan2b 600	. . 3 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
6	5	3adant1 1136	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
7	2, 6	impbid 213	1 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∃wex 1786  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-ex 1787  df-mo 2543  df-eu 2573

This theorem is referenced by: (None)