MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm4.71 Structured version   Visualization version   GIF version

Theorem pm4.71 549
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 470 . . 3 ((𝜑𝜓) → 𝜑)
21biantru 521 . 2 ((𝜑 → (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
3 anclb 537 . 2 ((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
4 dfbi2 462 . 2 ((𝜑 ↔ (𝜑𝜓)) ↔ ((𝜑 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜑)))
52, 3, 43bitr4i 294 1 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384
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 198  df-an 385
This theorem is referenced by:  pm4.71r  550  pm4.71i  551  pm4.71d  553  bigolden  1042  pm5.75  1044  rabid2  3307  rabid2f  3308  dfss2  3786  disj3  4218  dmopab3  5538  mptfnf  6222  nanorxor  39001
  Copyright terms: Public domain W3C validator