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Theorem bj-sb 32319
Description: A weak variant of sbid2 2412 not requiring ax-13 2245 nor ax-10 2016. On top of Tarski's FOL, one implication requires only ax12v 2045, and the other requires only sp 2051. (Contributed by BJ, 25-May-2021.)
Assertion
Ref Expression
bj-sb (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-sb
StepHypRef Expression
1 ax12v 2045 . . . . 5 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21equcoms 1944 . . . 4 (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32com12 32 . . 3 (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
43alrimiv 1852 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
5 sp 2051 . . . . . . 7 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
65com12 32 . . . . . 6 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
76equcoms 1944 . . . . 5 (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
87a2i 14 . . . 4 ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥𝜑))
98alimi 1736 . . 3 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥𝜑))
10 bj-eqs 32305 . . 3 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥𝜑))
119, 10sylibr 224 . 2 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → 𝜑)
124, 11impbii 199 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
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  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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