Theorem eqnetrrid 3008

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 3001	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ne 2934

This theorem is referenced by:  xpcoidgend  14910  fclsfnflim  23983  ptcmplem2  24009  vieta1lem1  26286  vieta1lem2  26287  fsuppcurry1  32813  fsuppcurry2  32814  constrresqrtcl  33954  signsvfpn  34762  signsvfnn  34763  finxpreclem2  37642  finxp1o  37644  cdleme3h  40608  cdleme7ga  40621  imo72b2lem0  44518  imo72b2lem1  44522  fourierdlem42  46504