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Theorem pm4.71 369
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 102 . . 3 ((φ ψ) → φ)
21biantru 286 . 2 ((φ → (φ ψ)) ↔ ((φ → (φ ψ)) ((φ ψ) → φ)))
3 anclb 302 . 2 ((φψ) ↔ (φ → (φ ψ)))
4 dfbi2 368 . 2 ((φ ↔ (φ ψ)) ↔ ((φ → (φ ψ)) ((φ ψ) → φ)))
52, 3, 43bitr4i 201 1 ((φψ) ↔ (φ ↔ (φ ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm4.71r  370  pm4.71i  371  pm4.71d  373  bigolden  861  pm5.75  868  exintrbi  1521  rabid2  2480  dfss2  2928  disj3  3266  dmopab3  4491
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