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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 = 𝑦𝜑))

Proof of Theorem sb1
StepHypRef Expression
1 df-sb 2049 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simprbi 478 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class 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
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