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Theorem ifeq2da 4560
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
ifeq2da.1 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
Assertion
Ref Expression
ifeq2da (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2da
StepHypRef Expression
1 iftrue 4534 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2 iftrue 4534 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
31, 2eqtr4d 2774 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
43adantl 481 . 2 ((𝜑𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5 ifeq2da.1 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
65ifeq2d 4548 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
74, 6pm2.61dan 810 1 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  ifcif 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-un 3953  df-if 4529
This theorem is referenced by:  ifeq12da  4561  dfac12lem1  10144  ttukeylem3  10512  xmulcom  13252  xmulneg1  13255  subgmulg  19063  1marepvmarrepid  22397  pcopt2  24870  limcdif  25725  limcmpt  25732  limcres  25735  limccnp  25740  radcnv0  26267  leibpi  26788  efrlim  26815  dchrvmasumiflem2  27348  padicabvf  27477  padicabvcxp  27478  itg2addnclem  37003  fourierdlem73  45354  fourierdlem76  45357  fourierdlem89  45370  fourierdlem91  45372
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