ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3netr4d GIF version

Theorem 3netr4d 2282
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
3netr4d.1 (𝜑𝐴𝐵)
3netr4d.2 (𝜑𝐶 = 𝐴)
3netr4d.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3netr4d (𝜑𝐶𝐷)

Proof of Theorem 3netr4d
StepHypRef Expression
1 3netr4d.1 . 2 (𝜑𝐴𝐵)
2 3netr4d.2 . . 3 (𝜑𝐶 = 𝐴)
3 3netr4d.3 . . 3 (𝜑𝐷 = 𝐵)
42, 3neeq12d 2269 . 2 (𝜑 → (𝐶𝐷𝐴𝐵))
51, 4mpbird 165 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wne 2249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-ne 2250
This theorem is referenced by:  modsumfzodifsn  9530
  Copyright terms: Public domain W3C validator