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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axc11v | Structured version Visualization version GIF version | ||
| Description: Version of axc11 2434 with a disjoint variable condition, which does not require ax-13 2376 nor ax-10 2140. Remark: the following theorems (hbae 2435, nfae 2437, hbnae 2436, nfnae 2438, hbnaes 2439) would need to be totally unbundled to be proved without ax-13 2376, hence would be simple consequences of ax-5 1909 or nfv 1913. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-axc11v | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | axc11rv 2264 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | |
| 2 | 1 | bj-aecomsv 36810 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: (None) | 
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