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Theorem ibibr 356
Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
Assertion
Ref Expression
ibibr ((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Proof of Theorem ibibr
StepHypRef Expression
1 pm5.501 354 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 bicom 210 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
31, 2syl6bb 274 . 2 (𝜑 → (𝜓 ↔ (𝜓𝜑)))
43pm5.74i 258 1 ((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195
This theorem is referenced by:  tbt  357  rabxfrd  4806  ufileu  21471  abnotbtaxb  39531
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