ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifbieq1d GIF version

Theorem ifbieq1d 3462
Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
Hypotheses
Ref Expression
ifbieq1d.1 (𝜑 → (𝜓𝜒))
ifbieq1d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ifbieq1d (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))

Proof of Theorem ifbieq1d
StepHypRef Expression
1 ifbieq1d.1 . . 3 (𝜑 → (𝜓𝜒))
21ifbid 3461 . 2 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐴, 𝐶))
3 ifbieq1d.2 . . 3 (𝜑𝐴 = 𝐵)
43ifeq1d 3457 . 2 (𝜑 → if(𝜒, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
52, 4eqtrd 2148 1 (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  ifcif 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-un 3043  df-if 3443
This theorem is referenced by:  ctssdclemn0  6961  ctssdc  6964  enumctlemm  6965  iseqf1olemfvp  10210  seq3f1olemqsum  10213  seq3f1oleml  10216  seq3f1o  10217  bcval  10435  sumrbdclem  11085  summodclem3  11089  summodclem2a  11090  summodc  11092  zsumdc  11093  fsum3  11096  isumss  11100  isumss2  11102  fsum3cvg2  11103  fsum3ser  11106  fsumcl2lem  11107  fsumadd  11115  sumsnf  11118  fsummulc2  11157  isumlessdc  11205
  Copyright terms: Public domain W3C validator