Theorem eqnetrrid 3034

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 3027	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ne 2960

This theorem is used by:  xpcoidgend  15171  fclsfnflim  24401  ptcmplem2  24427  vieta1lem1  26688  vieta1lem2  26689  fsuppcurry1  33412  fsuppcurry2  33413  dflringlem3  34132  dflring4  34134  constrresqrtcl  34520  signsvfpn  35325  signsvfnn  35326  finxpreclem2  38581  finxp1o  38583  cdleme3h  41558  cdleme7ga  41571  imo72b2lem0  45426  imo72b2lem1  45430  fourierdlem42  47432