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  12846  starvndxnplusgndx  12847  starvndxnmulrndx  12848  scandxnbasendx  12858  scandxnplusgndx  12859  scandxnmulrndx  12860  vscandxnbasendx  12863  vscandxnplusgndx  12864  vscandxnmulrndx  12865  vscandxnscandx  12866  ipndxnbasendx  12876  ipndxnplusgndx  12877  ipndxnmulrndx  12878  slotsdifipndx  12879  tsetndxnplusgndx  12896  tsetndxnmulrndx  12897  tsetndxnstarvndx  12898  slotstnscsi  12899  plendxnplusgndx  12910  plendxnmulrndx  12911  plendxnscandx  12912  plendxnvscandx  12913  slotsdifplendx  12914  basendxnocndx  12917  plendxnocndx  12918  dsndxnplusgndx  12925  dsndxnmulrndx  12926  slotsdnscsi  12927  dsndxntsetndx  12928  slotsdifdsndx  12929  unifndxntsetndx  12935  slotsdifunifndx  12936  setsmsbasg  14823  setsmsdsg  14824