Metamath Proof Explorer


Theorem reupick3

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

Ref Expression
Assertion reupick3 ( ( ∃! 𝑥𝐴 𝜑 ∧ ∃ 𝑥𝐴 ( 𝜑𝜓 ) ∧ 𝑥𝐴 ) → ( 𝜑𝜓 ) )

Proof

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