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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sb | Structured version Visualization version GIF version |
Description: A weak variant of sbid2 2475 not requiring ax-13 2302 nor ax-10 2080. On top of Tarski's FOL, one implication requires only ax12v 2108, and the other requires only sp 2112. (Contributed by BJ, 25-May-2021.) |
Ref | Expression |
---|---|
bj-sb | ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12v 2108 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | 1 | equcoms 1978 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | 2 | com12 32 | . . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | 3 | alrimiv 1887 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | sp 2112 | . . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
6 | 5 | com12 32 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
7 | 6 | equcoms 1978 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
8 | 7 | a2i 14 | . . . 4 ⊢ ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑥 → 𝜑)) |
9 | 8 | alimi 1775 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑥 → 𝜑)) |
10 | bj-eqs 33551 | . . 3 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑)) | |
11 | 9, 10 | sylibr 226 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → 𝜑) |
12 | 4, 11 | impbii 201 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-12 2107 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 |
This theorem is referenced by: (None) |
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