Theorem eqnetrrid 3032

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 3025	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2957

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 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-ne 2958

This theorem is used by:  xpcoidgend  15052  fclsfnflim  24259  ptcmplem2  24285  vieta1lem1  26549  vieta1lem2  26550  fsuppcurry1  33203  fsuppcurry2  33204  dflringlem3  33914  dflring4  33916  constrresqrtcl  34295  signsvfpn  35101  signsvfnn  35102  finxpreclem2  38152  finxp1o  38154  cdleme3h  41116  cdleme7ga  41129  imo72b2lem0  45013  imo72b2lem1  45017  fourierdlem42  46985