Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifeq12da Structured version   Visualization version   GIF version

Theorem ifeq12da 4453
 Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
Hypotheses
Ref Expression
ifeq12da.1 ((𝜑𝜓) → 𝐴 = 𝐶)
ifeq12da.2 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
Assertion
Ref Expression
ifeq12da (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Proof of Theorem ifeq12da
StepHypRef Expression
1 ifeq12da.1 . . . 4 ((𝜑𝜓) → 𝐴 = 𝐶)
21ifeq1da 4451 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3 iftrue 4426 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4 iftrue 4426 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
53, 4eqtr4d 2796 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
62, 5sylan9eq 2813 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7 ifeq12da.2 . . . 4 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
87ifeq2da 4452 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9 iffalse 4429 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10 iffalse 4429 . . . 4 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
119, 10eqtr4d 2796 . . 3 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
128, 11sylan9eq 2813 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
136, 12pm2.61dan 812 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ifcif 4420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-un 3863  df-if 4421 This theorem is referenced by:  ifbieq12d2  4454  copco  23719  pcohtpylem  23720  rpvmasum2  26195  prjspnfv01  39980
 Copyright terms: Public domain W3C validator