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Theorem sb1 1940
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2458) or a non-freeness hypothesis (sb5f 2414). (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 1938 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simprbi 479 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744  [wsb 1937
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 385  df-sb 1938
This theorem is referenced by:  spsbe  1941  sb4  2384  sb4a  2385  sb4e  2390  sb6  2457  bj-sb4v  32882  bj-sb6  32892  bj-sb3b  32929  wl-sb5nae  33470
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