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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dveeq1-o16 | Structured version Visualization version GIF version | ||
| Description: Version of dveeq1 2385 using ax-c16 38893 instead of ax-5 1910. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1910. (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| dveeq1-o16 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax5eq 38933 | . 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧) | |
| 2 | ax5eq 38933 | . 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧) | |
| 3 | equequ1 2024 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧)) | |
| 4 | 1, 2, 3 | dvelimh 2455 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-c9 38891 ax-c16 38893 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: (None) | 
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