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Theorem bj-sbex 31815
 Description: If a proposition is true for a specific instance, then there exists an instance such that it is true for it. See spsbe 1871. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-sbex ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)

Proof of Theorem bj-sbex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ssb 31809 . . 3 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 ax6ev 1877 . . . 4 𝑦 𝑦 = 𝑡
3 exim 1751 . . . 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 206 . 2 ([𝑡/𝑥]b𝜑 → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
6 exim 1751 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
76eximi 1752 . 2 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
8 ax6ev 1877 . . . 4 𝑥 𝑥 = 𝑦
9 pm2.27 41 . . . 4 (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑))
108, 9ax-mp 5 . . 3 ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
1110exlimiv 1845 . 2 (∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
125, 7, 113syl 18 1 ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  [wssb 31808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-ssb 31809 This theorem is referenced by:  bj-ssbft  31831
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