Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl2imc Structured version   Visualization version   GIF version

Theorem syl2imc 41
 Description: A commuted version of syl2im 40. Implication-only version of syl2anr 496. (Contributed by BJ, 20-Oct-2021.)
Hypotheses
Ref Expression
syl2im.1 (𝜑𝜓)
syl2im.2 (𝜒𝜃)
syl2im.3 (𝜓 → (𝜃𝜏))
Assertion
Ref Expression
syl2imc (𝜒 → (𝜑𝜏))

Proof of Theorem syl2imc
StepHypRef Expression
1 syl2im.1 . . 3 (𝜑𝜓)
2 syl2im.2 . . 3 (𝜒𝜃)
3 syl2im.3 . . 3 (𝜓 → (𝜃𝜏))
41, 2, 3syl2im 40 . 2 (𝜑 → (𝜒𝜏))
54com12 32 1 (𝜒 → (𝜑𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7 This theorem is referenced by:  triun  4906  undifixp  8098  rankpwi  8847  2cshwcshw  13742  incexclem  14738  sumeven  15283  cygth  20093  cnpco  21244  txkgen  21628  numclwlk1lem2f1  27487  ontgval  32707  bj-dvelimdv1  33112  eel12131  39409  2ffzoeq  41817  iccpartgt  41842  bgoldbtbndlem3  42174
 Copyright terms: Public domain W3C validator