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Metamath Proof Explorer

Theorem pm14.12

Description: Theorem *14.12 in WhiteheadRussell p. 184. (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion pm14.12 ( ∃! 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )

Proof

Step Hyp Ref Expression
1 eumo ⊢ ( ∃! 𝑥 𝜑 → ∃* 𝑥 𝜑 )
2 nfv ⊢ Ⅎ 𝑦 𝜑
3 2 mo3 ⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
4 sbsbc ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
5 4 anbi2i ⊢ ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) )
6 5 imbi1i ⊢ ( ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
7 6 2albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
8 3 7 bitri ⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
9 1 8 sylib ⊢ ( ∃! 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )