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Metamath Proof Explorer

Theorem iota4an

Description: Theorem *14.23 in WhiteheadRussell p. 191. (Contributed by Andrew Salmon, 12-Jul-2011)

Ref Expression
Assertion iota4an ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 iota4 ⊢ ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) )
2 iotaex ⊢ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V
3 simpl ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
4 3 sbcth ⊢ ( ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) )
5 2 4 ax-mp ⊢ [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
6 sbcimg ⊢ ( ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V → ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ↔ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 ) ) )
7 2 6 ax-mp ⊢ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ↔ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 ) )
8 5 7 mpbi ⊢ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )
9 1 8 syl ⊢ ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )