Theorem spsbe 1871

Description: A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Assertion

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

Proof of Theorem spsbe

Step	Hyp	Ref	Expression
1		sb1 1870	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		exsimpr 1784	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
3	1, 2	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868

This theorem is referenced by:  sbft  2367  2mo  2539  bj-sbftv  31951  bj-sbfvv  31953  wl-lem-moexsb  32529  spsbce-2  37602  sb5ALT  37752  sb5ALTVD  38171