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Theorem ralnex2OLD 3258
Description: Obsolete version of ralnex2 3257 as of 18-May-2023. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ralnex2OLD (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem ralnex2OLD
StepHypRef Expression
1 notnotb 316 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
2 notnotb 316 . . . 4 (𝜑 ↔ ¬ ¬ 𝜑)
322rexbii 3245 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑)
4 rexnal2 3255 . . 3 (∃𝑥𝐴𝑦𝐵 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑)
53, 4bitr2i 277 . 2 (¬ ∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜑)
61, 5xchbinx 335 1 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wral 3135  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-ral 3140  df-rex 3141
This theorem is referenced by: (None)
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