Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  spsbe Structured version   Visualization version   GIF version

Theorem spsbe 2033
 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 2016. (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 2016 . . 3 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 ax6ev 1930 . . . 4 𝑦 𝑦 = 𝑡
3 exim 1796 . . . 4 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (∃𝑦 𝑦 = 𝑡 → ∃𝑦𝑥(𝑥 = 𝑦𝜑)))
42, 3mpi 20 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
51, 4sylbi 209 . 2 ([𝑡 / 𝑥]𝜑 → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
6 exsbim 1958 . 2 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
75, 6syl 17 1 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1505  ∃wex 1742  [wsb 2015 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928 This theorem depends on definitions:  df-bi 199  df-ex 1743  df-sb 2016 This theorem is referenced by:  sbv  2040  sbft  2198  2mo  2680  noel  4184  bj-ssbft  33500  wl-lem-moexsb  34242  nsb  38542  spsbce-2  40126  sb5ALT  40275  sb5ALTVD  40663
 Copyright terms: Public domain W3C validator