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Metamath Proof Explorer

Theorem dfsb7

Description: An alternate definition of proper substitution df-sb . By introducing a dummy variable y in the definiens, we are able to eliminate any distinct variable restrictions among the variables t , x , and ph of the definiendum. No distinct variable conflicts arise because y effectively insulates t from x . To achieve this, we use a chain of two substitutions in the form of sb5 , first y for x then t for y . Compare Definition 2.1'' of Quine p. 17, which is obtained from this theorem by applying df-clab . Theorem sb7h provides a version where ph and y don't have to be distinct. (Contributed by NM, 28-Jan-2004) Revise df-sb . (Revised by BJ, 25-Dec-2020) (Proof shortened by Wolf Lammen, 3-Sep-2023)

Ref Expression
Assertion dfsb7 ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑡 ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 sbalex ⊢ ( ∃ 𝑦 ( 𝑦 = 𝑡 ∧ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2 sbalex ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
3 2 anbi2i ⊢ ( ( 𝑦 = 𝑡 ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ↔ ( 𝑦 = 𝑡 ∧ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
4 3 exbii ⊢ ( ∃ 𝑦 ( 𝑦 = 𝑡 ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ↔ ∃ 𝑦 ( 𝑦 = 𝑡 ∧ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
5 dfsb ⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
6 1 4 5 3bitr4ri ⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑡 ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )