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Metamath Proof Explorer

Theorem iota4

Description: Theorem *14.22 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 12-Jul-2011)

Ref Expression
Assertion iota4 ( ∃! 𝑥 𝜑 → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 eu6 ⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) )
2 biimpr ⊢ ( ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( 𝑥 = 𝑧 → 𝜑 ) )
3 2 alimi ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) )
4 sb6 ⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) )
5 3 4 sylibr ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ 𝑧 / 𝑥 ] 𝜑 )
6 iotaval ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( ℩ 𝑥 𝜑 ) = 𝑧 )
7 6 eqcomd ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → 𝑧 = ( ℩ 𝑥 𝜑 ) )
8 dfsbcq2 ⊢ ( 𝑧 = ( ℩ 𝑥 𝜑 ) → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 ) )
9 7 8 syl ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 ) )
10 5 9 mpbid ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )
11 10 exlimiv ⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )
12 1 11 sylbi ⊢ ( ∃! 𝑥 𝜑 → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )