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Theorem pm4.71 386
 Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
Assertion
Ref Expression
pm4.71 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))

Proof of Theorem pm4.71
StepHypRef Expression
1 simpl 108 . . 3 ((𝜑𝜓) → 𝜑)
21biantru 300 . 2 ((𝜑 → (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
3 anclb 317 . 2 ((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
4 dfbi2 385 . 2 ((𝜑 ↔ (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
52, 3, 43bitr4i 211 1 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  pm4.71r  387  pm4.71i  388  pm4.71d  390  bigolden  939  pm5.75  946  exintrbi  1612  rabid2  2607  dfss2  3086  disj3  3415  dmopab3  4752
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