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Theorem spsbe 1723
Description: A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
spsbe ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbe
StepHypRef Expression
1 sb1 1649 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 simpr 103 . . 3 ((𝑥 = 𝑦𝜑) → 𝜑)
32eximi 1491 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
41, 3syl 14 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wex 1381  [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  sbft  1728
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