Theorem com13 80

Description: Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

Hypothesis

Ref	Expression
com3.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))

Assertion

Ref	Expression
com13	⊢ (𝜒 → (𝜓 → (𝜑 → 𝜃)))

Proof of Theorem com13

Step	Hyp	Ref	Expression
1		com3.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2	1	com3r 79	. 2 ⊢ (𝜒 → (𝜑 → (𝜓 → 𝜃)))
3	2	com23 78	1 ⊢ (𝜒 → (𝜓 → (𝜑 → 𝜃)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  com24  87  an13s  573  an31s  576  3imp31  1227  3imp21  1229  funopg  5411  f1o2ndf1  6464  brecop  6899  fiintim  7238  elpq  10060  xnn0lenn0nn0  10278  elfz0ubfz0  10543  elfz0fzfz0  10544  fz0fzelfz0  10545  fz0fzdiffz0  10548  fzo1fzo0n0  10606  elfzodifsumelfzo  10630  ssfzo12  10653  ssfzo12bi  10654  facwordi  11194  fihashf1rn  11243  swrdswrdlem  11492  swrdswrd  11493  wrd2ind  11511  swrdccatin1  11513  pfxccatin12lem2  11519  swrdccat  11523  reuccatpfxs1lem  11534  oddnn02np1  12666  oddge22np1  12667  evennn02n  12668  evennn2n  12669  dfgcd2  12810  sqrt2irr  12960  lmodfopnelem1  14745  mpomulcn  15758  zabsle1  16289  gausslemma2dlem1a  16348  2lgsoddprm  16403  upgredg2vtx  16560  usgruspgrben  16598  usgredg2vlem2  16635  edg0usgr  16659  uspgr2wlkeq  16777  clwwlkn1loopb  16832  clwwlkext2edg  16834  clwwlknonex2lem2  16850  bj-inf2vnlem2  17168