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Theorem spsbeOLD 2089
Description: Obsolete version of spsbe 2088 as of 11-Jul-2023. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
spsbeOLD ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbeOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2070 . . 3 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 ax6ev 1972 . . . 4 𝑦 𝑦 = 𝑡
3 exim 1835 . . . 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 220 . 2 ([𝑡 / 𝑥]𝜑 → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
6 exim 1835 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
76eximi 1836 . 2 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
8 ax6ev 1972 . . . 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 1931 . 2 (∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
125, 7, 113syl 18 1 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-sb 2070
This theorem is referenced by: (None)
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