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

Theorem ifpn 1074
Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)
Assertion
Ref Expression
ifpn (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Proof of Theorem ifpn
StepHypRef Expression
1 ancom 464 . 2 (((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜒) ∧ (¬ 𝜑𝜓)))
2 dfifp5 1068 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
3 dfifp3 1066 . 2 (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑𝜒) ∧ (¬ 𝜑𝜓)))
41, 2, 33bitr4i 306 1 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847  if-wif 1063
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 210  df-an 400  df-or 848  df-ifp 1064
This theorem is referenced by:  ifpfal  1077  wl-ifp-ncond2  35373  wl-df3xor2  35377  wl-2xor  35391  wl-df3maxtru1  35400  ifpxorcor  40768
  Copyright terms: Public domain W3C validator