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Theorem pm4.71 382
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 297 . 2 ((𝜑 → (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
3 anclb 313 . 2 ((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
4 dfbi2 381 . 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-1 5  ax-2 6  ax-mp 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  383  pm4.71i  384  pm4.71d  386  bigolden  902  pm5.75  909  exintrbi  1570  rabid2  2544  dfss2  3015  disj3  3339  dmopab3  4662
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