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Theorem iftrued 3445
Description: Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
iftrued.1 (𝜑𝜒)
Assertion
Ref Expression
iftrued (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)

Proof of Theorem iftrued
StepHypRef Expression
1 iftrued.1 . 2 (𝜑𝜒)
2 iftrue 3443 . 2 (𝜒 → if(𝜒, 𝐴, 𝐵) = 𝐴)
31, 2syl 14 1 (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1312  ifcif 3438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-if 3439
This theorem is referenced by:  eqifdc  3470  mposnif  5817  fimax2gtrilemstep  6745  updjudhcoinlf  6915  omp1eomlem  6929  difinfsnlem  6934  ctssdclemn0  6945  ctssdc  6948  enumctlemm  6949  fodju0  6967  iseqf1olemnab  10148  iseqf1olemab  10149  iseqf1olemqk  10154  iseqf1olemfvp  10157  seq3f1olemqsumkj  10158  seq3f1olemqsum  10160  seq3f1oleml  10163  seq3f1o  10164  fser0const  10176  expnnval  10183  2zsupmax  10883  xrmaxifle  10901  xrmaxiflemab  10902  xrmaxiflemlub  10903  xrmaxiflemcom  10904  summodclem3  11035  summodclem2a  11036  isum  11040  fsum3  11042  isumss  11046  fsumcl2lem  11053  fsumadd  11061  fsummulc2  11103  cvgratz  11187  ef0lem  11211  gcdval  11490  ennnfonelemss  11762  ennnfonelemkh  11764  ennnfonelemhf1o  11765  ressid2  11855  nninfalllemn  12883  nninfsellemeq  12891  nninfsellemeqinf  12893  nninffeq  12897
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