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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 ( ( ∃! 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 ) )