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Theorem pm4.71 547
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 468 . . 3 ((𝜑𝜓) → 𝜑)
21biantru 519 . 2 ((𝜑 → (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
3 anclb 535 . 2 ((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
4 dfbi2 460 . 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 382
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 383
This theorem is referenced by:  pm4.71r  548  pm4.71i  549  pm4.71d  551  bigolden  1013  pm5.75  1015  rabid2  3267  rabid2f  3268  dfss2  3740  disj3  4165  dmopab3  5474  mptfnf  6154  nanorxor  39028
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