Theorem eqnetrd 2371

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqnetrd.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
eqnetrd	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem eqnetrd

Step	Hyp	Ref	Expression
1		eqnetrd.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
2		eqnetrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	neeq1d 2365	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
4	1, 3	mpbird 167	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1353   ≠ wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348

This theorem is referenced by:  eqnetrrd  2373  frecabcl  6397  frecsuclem  6404  omp1eomlem  7090  xaddnemnf  9853  xaddnepnf  9854  hashprg  10781  bezoutr1  12026  phibndlem  12208  dfphi2  12212  lgsne0  14310  2sqlem8a  14329  2sqlem8  14330