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Theorem bj-sbex 32321
Description: If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 1881 which, due to the specific form of df-sb 1878, uses fewer axioms. (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 32315 . . 3 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 ax6ev 1887 . . . 4 𝑦 𝑦 = 𝑡
3 exim 1758 . . . 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 207 . 2 ([𝑡/𝑥]b𝜑 → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
6 exim 1758 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
76eximi 1759 . 2 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
8 ax6ev 1887 . . . 4 𝑥 𝑥 = 𝑦
9 pm2.27 42 . . . 4 (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑))
108, 9ax-mp 5 . . 3 ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
1110exlimiv 1855 . 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 1478  wex 1701  [wssb 32314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-ssb 32315
This theorem is referenced by:  bj-ssbft  32337
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