Theorem an12 561

Description: Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
an12	|- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) )

Proof of Theorem an12

Step	Hyp	Ref	Expression
1		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
2	1	anbi1i 458	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ps /\ ph ) /\ ch ) )
3		anass 401	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
4		anass 401	. 2 |- ( ( ( ps /\ ph ) /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) )
5	2, 3, 4	3bitr3i 210	1 |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) )

Syntax hints:   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  an32  562  an13  563  an12s  565  an4  586  ceqsrexv  2894  rmoan  2964  2reuswapdc  2968  reuind  2969  2rmorex  2970  sbccomlem  3064  elunirab  3852  rexxfrd  4498  opeliunxp  4718  elres  4982  resoprab  6018  ov6g  6061  opabex3d  6178  opabex3  6179  xpassen  6889  distrnqg  7454  distrnq0  7526  rexuz2  9655  2clim  11466  issubrg  13777  isbasis2g  14281  tgval2  14287