Theorem sb1 2051

Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2272) or a non-freeness hypothesis (sb5f 2532). (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 2049	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simprbi 478	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1851  [wsb 2048

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 383  df-sb 2049

This theorem is referenced by:  spsbe  2052  sb6  2271  sb3b  2501  sb4  2502  sb4a  2503  sb4e  2508  sb6OLD  2575  bj-sb4v  33087  bj-sb6  33097