Theorem neeq12i 2384

Description: Inference for inequality. (Contributed by NM, 24-Jul-2012.)

Hypotheses

Ref	Expression
neeq1i.1	⊢ 𝐴 = 𝐵
neeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
neeq12i	⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)

Proof of Theorem neeq12i

Step	Hyp	Ref	Expression
1		neeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
2	1	neeq2i 2383	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐴 ≠ 𝐷)
3		neeq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	neeq1i 2382	. 2 ⊢ (𝐴 ≠ 𝐷 ↔ 𝐵 ≠ 𝐷)
5	2, 4	bitri 184	1 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)

Syntax hints:   ↔ wb 105   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  3netr3g  2401  3netr4g  2402  starvndxnbasendx  12819  starvndxnplusgndx  12820  starvndxnmulrndx  12821  scandxnbasendx  12831  scandxnplusgndx  12832  scandxnmulrndx  12833  vscandxnbasendx  12836  vscandxnplusgndx  12837  vscandxnmulrndx  12838  vscandxnscandx  12839  ipndxnbasendx  12849  ipndxnplusgndx  12850  ipndxnmulrndx  12851  slotsdifipndx  12852  tsetndxnplusgndx  12869  tsetndxnmulrndx  12870  tsetndxnstarvndx  12871  slotstnscsi  12872  plendxnplusgndx  12883  plendxnmulrndx  12884  plendxnscandx  12885  plendxnvscandx  12886  slotsdifplendx  12887  dsndxnplusgndx  12894  dsndxnmulrndx  12895  slotsdnscsi  12896  dsndxntsetndx  12897  slotsdifdsndx  12898  unifndxntsetndx  12904  slotsdifunifndx  12905  setsmsbasg  14715  setsmsdsg  14716