Theorem 2thd 175

Description: Two truths are equivalent (deduction form). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.)

Hypotheses

Ref	Expression
2thd.1	|- ( ph -> ps )
2thd.2	|- ( ph -> ch )

Assertion

Ref	Expression
2thd	|- ( ph -> ( ps <-> ch ) )

Proof of Theorem 2thd

Step	Hyp	Ref	Expression
1		2thd.1	. 2 |- ( ph -> ps )
2		2thd.2	. 2 |- ( ph -> ch )
3		pm5.1im 173	. 2 |- ( ps -> ( ch -> ( ps <-> ch ) ) )
4	1, 2, 3	sylc 62	1 |- ( ph -> ( ps <-> ch ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  biort  841  rspcime  2937  abvor0dc  3545  exmidsssn  4339  euotd  4395  nn0eln0  4767  elabrex  5963  elabrexg  5964  riota5f  6065  nntri3  6770  modom  7108  fin0  7189  2omap  7318  omp1eomlem  7434  ctssdccl  7451  ismkvnex  7495  finacn  7560  acnccim  7638  nn1m1nn  9324  xrlttri3  10209  nltpnft  10226  ngtmnft  10229  xrrebnd  10231  xltadd1  10288  xsubge0  10293  xposdif  10294  xlesubadd  10295  xleaddadd  10299  iccshftr  10406  iccshftl  10408  iccdil  10410  icccntr  10412  fzaddel  10475  elfzomelpfzo  10659  xqltnle  10712  nnesq  11110  nn0sqdc  11160  hashnncl  11248  zfz1isolemiso  11305  swrdspsleq  11453  mod2eq1n2dvds  12662  m1exp1  12684  dfgcd3  12803  dvdssq  12824  pcdvdsb  13119  pceq0  13121  issubg3  14044  lmss  15396  lmres  15398  eupth2lem3lem6fi  16810