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Mirrors > Home > MPE Home > Th. List > Mathboxes > spALT | Structured version Visualization version GIF version |
Description: sp 2179 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2179 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
spALT | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (∀𝑥𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜑)) | |
2 | 1 | axc4i 2319 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) |
3 | axc10 2386 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | |
4 | 2, 3 | syl 17 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-10 2140 ax-12 2174 ax-13 2373 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ex 1786 df-nf 1790 |
This theorem is referenced by: (None) |
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