Theorem sb1 1883

Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2430) or a non-freeness hypothesis (sb5f 2386). (Contributed by NM, 13-May-1993.)

Assertion

Ref	Expression
sb1	⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem sb1

Step	Hyp	Ref	Expression
1		df-sb 1881	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simprbi 480	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704  [wsb 1880

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 197  df-an 386  df-sb 1881

This theorem is referenced by:  spsbe  1884  sb4  2356  sb4a  2357  sb4e  2362  sb6  2429  bj-sb4v  32757  bj-sb6  32767  bj-sb3b  32804  wl-sb5nae  33340