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Metamath Proof Explorer

Theorem mob2

Description: Consequence of "at most one". (Contributed by NM, 2-Jan-2015)

Ref Expression
Hypothesis moi2.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion mob2 ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ∧ 𝜑 ) → ( 𝑥 = 𝐴 ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 moi2.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2 simp3 ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ∧ 𝜑 ) → 𝜑 )
3 2 1 syl5ibcom ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ∧ 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
4 nfv ⊢ Ⅎ 𝑥 𝜓
5 4 1 sbhypf ⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
6 5 anbi2d ⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )
7 eqeq2 ⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) )
8 6 7 imbi12d ⊢ ( 𝑦 = 𝐴 → ( ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝐴 ) ) )
9 8 spcgv ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) → ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝐴 ) ) )
10 nfs1v ⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
11 sbequ12 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
12 10 11 mo4f ⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
13 sp ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) → ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
14 12 13 sylbi ⊢ ( ∃* 𝑥 𝜑 → ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
15 9 14 impel ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ) → ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝐴 ) )
16 15 expd ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ) → ( 𝜑 → ( 𝜓 → 𝑥 = 𝐴 ) ) )
17 16 3impia ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ∧ 𝜑 ) → ( 𝜓 → 𝑥 = 𝐴 ) )
18 3 17 impbid ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∃* 𝑥 𝜑 ∧ 𝜑 ) → ( 𝑥 = 𝐴 ↔ 𝜓 ) )