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Mirrors > Home > MPE Home > Th. List > dtrucor | Structured version Visualization version GIF version |
Description: Corollary of dtru 5273. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 5275. (Contributed by NM, 27-Jun-2002.) |
Ref | Expression |
---|---|
dtrucor.1 | ⊢ 𝑥 = 𝑦 |
Ref | Expression |
---|---|
dtrucor | ⊢ 𝑥 ≠ 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtru 5273 | . . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | 1 | pm2.21i 119 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 ≠ 𝑦) |
3 | dtrucor.1 | . 2 ⊢ 𝑥 = 𝑦 | |
4 | 2, 3 | mpg 1798 | 1 ⊢ 𝑥 ≠ 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1535 ≠ wne 3018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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