Theorem sb4e

Description: One direction of a simplified definition of substitution that unlike sb4b does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	sb4e	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb1	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
2		equs5e	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )
3	1 2	syl	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )