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Theorem 3netr3d 2281
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
3netr3d.1 (𝜑𝐴𝐵)
3netr3d.2 (𝜑𝐴 = 𝐶)
3netr3d.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
3netr3d (𝜑𝐶𝐷)

Proof of Theorem 3netr3d
StepHypRef Expression
1 3netr3d.1 . 2 (𝜑𝐴𝐵)
2 3netr3d.2 . . 3 (𝜑𝐴 = 𝐶)
3 3netr3d.3 . . 3 (𝜑𝐵 = 𝐷)
42, 3neeq12d 2269 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
51, 4mpbid 145 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: (None)
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