Theorem eupickb 2635

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 2633	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
2	1	3adant2 1131	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3		exancom 1861	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4		eupick 2633	. . . 4 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓 ∧ 𝜑)) → (𝜓 → 𝜑))
5	3, 4	sylan2b 594	. . 3 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
6	5	3adant1 1130	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
7	2, 6	impbid 212	1 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∃wex 1779  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)