Theorem necomd 2433

Description: Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Hypothesis

Ref	Expression
necomd.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
necomd	⊢ (𝜑 → 𝐵 ≠ 𝐴)

Proof of Theorem necomd

Step	Hyp	Ref	Expression
1		necomd.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		necom 2431	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝜑 → 𝐵 ≠ 𝐴)

Syntax hints:   → wi 4   ≠ wne 2347

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348

This theorem is referenced by:  difsnb  3737  0nelop  4250  frecabcl  6402  fidifsnen  6872  tpfidisj  6929  omp1eomlem  7095  difinfsnlem  7100  fodjuomnilemdc  7144  en2eleq  7196  en2other2  7197  netap  7255  2omotaplemap  7258  ltned  8073  lt0ne0  8387  zdceq  9330  zneo  9356  xrlttri3  9799  qdceq  10249  flqltnz  10289  nn0opthd  10704  hashdifpr  10802  sumtp  11424  isprm2lem  12118  oddprm  12261  pcmpt  12343  ennnfonelemex  12417  lgsneg  14510