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Theorem equs3 1872
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.)
Assertion
Ref Expression
equs3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))

Proof of Theorem equs3
StepHypRef Expression
1 alinexa 1767 . 2 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
21con2bii 347 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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