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Theorem frege54cor0a 38576
Description: Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege54cor0a ((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Proof of Theorem frege54cor0a
StepHypRef Expression
1 ax-frege28 38543 . . . 4 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21anim2i 594 . . 3 (((𝜓𝜑) ∧ (𝜑𝜓)) → ((𝜓𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
3 con4 112 . . . 4 ((¬ 𝜓 → ¬ 𝜑) → (𝜑𝜓))
43anim2i 594 . . 3 (((𝜓𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)) → ((𝜓𝜑) ∧ (𝜑𝜓)))
52, 4impbii 199 . 2 (((𝜓𝜑) ∧ (𝜑𝜓)) ↔ ((𝜓𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
6 dfbi2 663 . 2 ((𝜓𝜑) ↔ ((𝜓𝜑) ∧ (𝜑𝜓)))
7 dfifp2 1052 . 2 (if-(𝜓, 𝜑, ¬ 𝜑) ↔ ((𝜓𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)))
85, 6, 73bitr4i 292 1 ((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  if-wif 1050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 38543
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051
This theorem is referenced by:  frege54cor1a  38577  frege55lem1a  38579  frege55lem2a  38580
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