Theorem sb1ALT 2582

Description: Alternate version of sb1 2503. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sb1ALT	⊢ (𝜃 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem sb1ALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. 2 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simprbi 499	1 ⊢ (𝜃 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

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

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

This theorem is referenced by:  sb4vOLDALT  2584  sb4ALT  2588  spsbeALT  2589  sb4aALT  2598