Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  infeq1d GIF version

Theorem infeq1d 6899
 Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypothesis
Ref Expression
infeq1d.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
infeq1d (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Proof of Theorem infeq1d
StepHypRef Expression
1 infeq1d.1 . 2 (𝜑𝐵 = 𝐶)
2 infeq1 6898 . 2 (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
31, 2syl 14 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  infcinf 6870 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-uni 3737  df-sup 6871  df-inf 6872 This theorem is referenced by:  xrbdtri  11057  lcmval  11755  lcmass  11777  bdmetval  12683  bdxmet  12684  qtopbasss  12704
 Copyright terms: Public domain W3C validator