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Theorem ralinexa 2345
 Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
ralinexa (x A (φ → ¬ ψ) ↔ ¬ x A (φ ψ))

Proof of Theorem ralinexa
StepHypRef Expression
1 imnan 623 . . 3 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
21ralbii 2324 . 2 (x A (φ → ¬ ψ) ↔ x A ¬ (φ ψ))
3 ralnex 2310 . 2 (x A ¬ (φ ψ) ↔ ¬ x A (φ ψ))
42, 3bitri 173 1 (x A (φ → ¬ ψ) ↔ ¬ x A (φ ψ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wral 2300  ∃wrex 2301 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-4 1397  ax-17 1416 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by: (None)
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