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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axc10v | Structured version Visualization version GIF version |
Description: Version of axc10 2394 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-axc10v | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6v 1962 | . . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | |
2 | con3 156 | . . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) | |
3 | 2 | al2imi 1807 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦)) |
4 | 1, 3 | mtoi 200 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
5 | axc7 2327 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | |
6 | 4, 5 | syl 17 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: (None) |
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