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Theorem pm4.71 665
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-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 474 . . 3 ((𝜑𝜓) → 𝜑)
21biantru 527 . 2 ((𝜑 → (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
3 anclb 571 . 2 ((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
4 dfbi2 663 . 2 ((𝜑 ↔ (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
52, 3, 43bitr4i 292 1 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383
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 197  df-an 385
This theorem is referenced by:  pm4.71r  666  pm4.71i  667  pm4.71d  669  bigolden  1014  pm5.75  1016  rabid2  3245  rabid2f  3246  dfss2  3720  disj3  4152  dmopab3  5480  mptfnf  6164  nanorxor  38975
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