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Theorem bj-sb 34078
Description: A weak variant of sbid2 2552 not requiring ax-13 2392 nor ax-10 2146. On top of Tarski's FOL, one implication requires only ax12v 2180, and the other requires only sp 2184. (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 2180 . . . . 5 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21equcoms 2028 . . . 4 (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32com12 32 . . 3 (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
43alrimiv 1929 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
5 sp 2184 . . . . . . 7 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
65com12 32 . . . . . 6 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
76equcoms 2028 . . . . 5 (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
87a2i 14 . . . 4 ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥𝜑))
98alimi 1813 . . 3 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥𝜑))
10 bj-eqs 34065 . . 3 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥𝜑))
119, 10sylibr 237 . 2 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → 𝜑)
124, 11impbii 212 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536
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 1912  ax-6 1971  ax-7 2016  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by: (None)
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