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Metamath Proof Explorer

Theorem eqeu

Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009)

Ref Expression
Hypothesis eqeu.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion eqeu ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) → ∃! 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 eqeu.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2 1 spcegv ⊢ ( 𝐴 ∈ 𝐵 → ( 𝜓 → ∃ 𝑥 𝜑 ) )
3 2 imp ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∃ 𝑥 𝜑 )
4 3 3adant3 ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) → ∃ 𝑥 𝜑 )
5 eqeq2 ⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) )
6 5 imbi2d ⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 → 𝑥 = 𝑦 ) ↔ ( 𝜑 → 𝑥 = 𝐴 ) ) )
7 6 albidv ⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) )
8 7 spcegv ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) )
9 8 imp ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )
10 9 3adant2 ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) → ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )
11 eu3v ⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) )
12 4 10 11 sylanbrc ⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ) → ∃! 𝑥 𝜑 )