Theorem sb3

Description: One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

	Ref	Expression
Assertion	sb3	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb3b	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )
2	1	biimprd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 ) )