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Mirrors > Home > MPE Home > Th. List > dveeq1 | Structured version Visualization version GIF version |
Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2146. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dveeq1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf1 2370 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | |
2 | 1 | nf5rd 2181 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: nfeqf 2372 axc11n 2417 |
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