Metamath Proof Explorer


Theorem sb3bOLD

Description: Obsolete version of sb3b as of 21-Feb-2024. (Contributed by BJ, 6-Oct-2018) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sb3bOLD ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 sb1 ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) )
2 sb3 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) )
3 1 2 impbid2 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )