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Theorem spsbe 1767
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 1693 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 simpr 108 . . 3 ((𝑥 = 𝑦𝜑) → 𝜑)
32eximi 1534 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
41, 3syl 14 1 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1424  [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-sb 1690
This theorem is referenced by:  sbft  1773
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