Theorem reupick3

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016)

	Ref	Expression
Assertion	reupick3	⊢ ( ( ∃! 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
2		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) )
3		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) )
4	3	exbii	⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) )
5	2 4	bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) )
6		eupick	⊢ ( ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
7	1 5 6	syl2anb	⊢ ( ( ∃! 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
8	7	expd	⊢ ( ( ∃! 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) → ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
9	8	3impia	⊢ ( ( ∃! 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝜑 → 𝜓 ) )