Theorem spsbeALT 2589

Description: Alternate version of spsbe 2088. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
spsbeALT	⊢ (𝜃 → ∃𝑥𝜑)

Proof of Theorem spsbeALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	sb1ALT 2582	. 2 ⊢ (𝜃 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3		exsimpr 1870	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
4	2, 3	syl 17	1 ⊢ (𝜃 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  sbftALT  2593