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Theorem ralnexOLD 2987
 Description: Obsolete proof of ralnex 2986 as of 16-Jul-2021. (Contributed by NM, 21-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ralnexOLD (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)

Proof of Theorem ralnexOLD
StepHypRef Expression
1 df-ral 2912 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 alinexa 1767 . . 3 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥𝐴𝜑))
3 df-rex 2913 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
42, 3xchbinxr 325 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥𝐴 𝜑)
51, 4bitri 264 1 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908 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  df-ral 2912  df-rex 2913 This theorem is referenced by: (None)
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