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Theorem bj-sb56 33596
 Description: Proof of sb56 2241 from Tarski, ax-10 2112 (modal5) and bj-ax12 33592. (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 33592 . . . 4 𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 pm3.31 450 . . . . 5 ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
32aleximi 1813 . . . 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 hbe1a 2115 . . 3 (∃𝑥𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl 17 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
7 equs4v 1983 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
86, 7impbii 210 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762 This theorem is referenced by: (None)
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