Assume that every object fails . A fresh existential witness for the premise then has both and its negation. Eliminate the witness before closing the assumption about all objects:
Table 1.
Eliminate the witness inside the negation subproof
.
LineOpen assumptionsFormulaJustification
11Premise
21, 2Temporary assumption
31, 2, 3Temporary assumption
41, 2, 3universal elimination 2; term
51, 2, 3negation elimination rule 3, 4
61, 2existential elimination 1, 3, 5; fresh
71negation introduction 2, 6
The conclusion is
(1)
The witness occurs neither in the conclusion of existential elimination nor in its other open assumptions, so the eigenvariable condition holds.

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