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Metamath Proof Explorer

Theorem iotasbc5

Description: Theorem *14.205 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion iotasbc5 ( ∃! 𝑥 𝜑 → ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 sbc5 ⊢ ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) )
2 1 a1i ⊢ ( ∃! 𝑥 𝜑 → ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) ) )