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Theorem bj-sb56 32764
Description: Proof of sb56 2188 from Tarski, ax-10 2059 (modal5) and bj-ax12 32759. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sb56 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-sb56
StepHypRef Expression
1 bj-ax12 32759 . . . 4 𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 pm3.31 460 . . . . 5 ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
32aleximi 1799 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝑥(𝑥 = 𝑦𝜑)))
41, 3ax-mp 5 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝑥(𝑥 = 𝑦𝜑))
5 bj-modal5e 32761 . . 3 (∃𝑥𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl 17 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
7 equs4v 1976 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
86, 7impbii 199 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-dfssb2  32765
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