Theorem spsbeOLDOLD 2510

Description: Obsolete version of spsbe 2087 as of 7-Jul-2023. A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
spsbeOLDOLD	⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbeOLDOLD

Step	Hyp	Ref	Expression
1		sb1 2502	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		exsimpr 1869	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
3	1, 2	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1779  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)