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Mirrors > Home > MPE Home > Th. List > Mathboxes > dveeq1-o16 | Structured version Visualization version GIF version |
Description: Version of dveeq1 2379 using ax-c16 37404 instead of ax-5 1914. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1914. (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 37444 | . 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧) | |
2 | ax5eq 37444 | . 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧) | |
3 | equequ1 2029 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧)) | |
4 | 1, 2, 3 | dvelimh 2449 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-c9 37402 ax-c16 37404 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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