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Theorem ralnexOLD 3022
Description: Obsolete proof of ralnex 3021 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 2946 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 alinexa 1810 . . 3 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥𝐴𝜑))
3 df-rex 2947 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
42, 3xchbinxr 324 . 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 383  wal 1521  wex 1744  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946  df-rex 2947
This theorem is referenced by: (None)
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