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Theorem pm4.71 611
 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 ↔ (φ ψ)))

Proof of Theorem pm4.71
StepHypRef Expression
1 simpl 443 . . 3 ((φ ψ) → φ)
21biantru 491 . 2 ((φ → (φ ψ)) ↔ ((φ → (φ ψ)) ((φ ψ) → φ)))
3 anclb 530 . 2 ((φψ) ↔ (φ → (φ ψ)))
4 dfbi2 609 . 2 ((φ ↔ (φ ψ)) ↔ ((φ → (φ ψ)) ((φ ψ) → φ)))
52, 3, 43bitr4i 268 1 ((φψ) ↔ (φ ↔ (φ ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358 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 177  df-an 360 This theorem is referenced by:  pm4.71r  612  pm4.71i  613  pm4.71d  615  bigolden  901  pm5.75  903  exintrbi  1613  rabid2  2788  dfss2  3262  disj3  3595  dmopab3  4917  resopab2  5001  resoprab2  5582
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