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 = 𝑦 ∧ 𝜑 ) ) )

Proof

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