Theorem eqnetrrid 3031

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 3024	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2956

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-ne 2957

This theorem is used by:  xpcoidgend  15121  fclsfnflim  24339  ptcmplem2  24365  vieta1lem1  26626  vieta1lem2  26627  fsuppcurry1  33309  fsuppcurry2  33310  dflringlem3  34021  dflring4  34023  constrresqrtcl  34402  signsvfpn  35207  signsvfnn  35208  finxpreclem2  38293  finxp1o  38295  cdleme3h  41272  cdleme7ga  41285  imo72b2lem0  45150  imo72b2lem1  45154  fourierdlem42  47128