Theorem spsbeALT

Description: Alternate version of spsbe . (Contributed by NM, 29-Jun-1993) (Proof shortened by Wolf Lammen, 3-May-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dfsb1.ph	⊢ ( 𝜃 ↔ ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )
Assertion	spsbeALT	⊢ ( 𝜃 → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		dfsb1.ph	⊢ ( 𝜃 ↔ ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )
2	1	sb1ALT	⊢ ( 𝜃 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
3		exsimpr	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑥 𝜑 )
4	2 3	syl	⊢ ( 𝜃 → ∃ 𝑥 𝜑 )