Theorem bj-sb 37174

Description: A weak variant of sbid2 2542 not requiring ax-13 2406 nor ax-10 2178. On top of Tarski's FOL, one implication requires only ax12v 2216, and the other requires only sp 2221. (Contributed by BJ, 25-May-2021.)

Assertion

Ref	Expression
bj-sb	⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-sb

Step	Hyp	Ref	Expression
1		ax12v 2216	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	equcoms 2043	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	com12 33	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	alrimiv 1950	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5		sp 2221	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
6	5	com12 33	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
7	6	equcoms 2043	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
8	7	a2i 15	. . . 4 ⊢ ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑥 → 𝜑))
9	8	alimi 1834	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑥 → 𝜑))
10		bj-eqs 37160	. . 3 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
11	9, 10	sylibr 237	. 2 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → 𝜑)
12	4, 11	impbii 212	1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by: (None)