Theorem eupickb

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

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

Proof

Step	Hyp	Ref	Expression
1		eupick	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜑 → 𝜓 ) )
2	1	3adant2	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃! 𝑥 𝜓 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜑 → 𝜓 ) )
3		exancom	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) )
4		eupick	⊢ ( ( ∃! 𝑥 𝜓 ∧ ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) ) → ( 𝜓 → 𝜑 ) )
5	3 4	sylan2b	⊢ ( ( ∃! 𝑥 𝜓 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜓 → 𝜑 ) )
6	5	3adant1	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃! 𝑥 𝜓 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜓 → 𝜑 ) )
7	2 6	impbid	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃! 𝑥 𝜓 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ( 𝜑 ↔ 𝜓 ) )