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Theorem spsbe 2088
 Description: Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.)
Assertion
Ref Expression
spsbe ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbe
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2071 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 alequexv 2008 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
3 exsbim 2009 . . 3 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
42, 3syl 17 . 2 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∃𝑥𝜑)
51, 4sylbi 220 1 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971 This theorem depends on definitions:  df-bi 210  df-ex 1783  df-sb 2071 This theorem is referenced by:  sbv  2096  sbft  2268  sb1  2493  2mo  2670  noel  4231  bj-sbft  34501  wl-lem-moexsb  35250  nsb  39690  spsbce-2  41459  sb5ALT  41605  sb5ALTVD  41993
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