Theorem eqnetrrid 3091

Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

Hypotheses

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

Assertion

Ref	Expression
eqnetrrid	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem eqnetrrid

Step	Hyp	Ref	Expression
1		eqnetrrid.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = 𝐴)
3		eqnetrrid.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
4	2, 3	eqnetrrd 3084	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1537   ≠ wne 3016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-ne 3017

This theorem is referenced by:  xpcoidgend  14335  fclsfnflim  22635  ptcmplem2  22661  vieta1lem1  24899  vieta1lem2  24900  fsuppcurry1  30461  fsuppcurry2  30462  signsvfpn  31855  signsvfnn  31856  finxpreclem2  34674  finxp1o  34676  cdleme3h  37386  cdleme7ga  37399  imo72b2lem1  40570  fourierdlem42  42483