Theorem sbequ12 1824

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sbequ12	|- ( x = y -> ( ph <-> [ y / x ] ph ) )

Proof of Theorem sbequ12

Step	Hyp	Ref	Expression
1		sbequ1 1821	. 2 |- ( x = y -> ( ph -> [ y / x ] ph ) )
2		sbequ2 1822	. 2 |- ( x = y -> ( [ y / x ] ph -> ph ) )
3	1, 2	impbid 129	1 |- ( x = y -> ( ph <-> [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 105   [wsb 1815

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-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563

This theorem depends on definitions:  df-bi 117  df-sb 1816

This theorem is referenced by:  sbequ12r  1825  sbequ12a  1826  sbid  1827  ax16  1866  sb8h  1907  sb8eh  1908  sb8  1909  sb8e  1910  ax16ALT  1912  sbco  2028  sbcomxyyz  2032  sb9v  2038  sb6a  2048  mopick  2165  clelab  2366  sbab  2368  nfabdw  2411  cbvralf  2777  cbvrexf  2778  cbvralsv  2802  cbvrexsv  2803  cbvrab  2819  sbhypf  2872  mob2  3006  reu2  3014  reu6  3015  sbcralt  3128  sbcrext  3129  sbcralg  3130  sbcreug  3132  cbvreucsf  3212  cbvrabcsf  3213  cbvopab1  4199  cbvopab1s  4201  csbopabg  4204  cbvmptf  4220  cbvmpt  4221  opelopabsb  4397  frind  4492  tfis  4725  findes  4745  opeliunxp  4825  ralxpf  4921  rexxpf  4922  cbviota  5337  csbiotag  5365  cbvriota  6040  csbriotag  6042  abrexex2g  6339  opabex3d  6340  opabex3  6341  abrexex2  6343  dfoprab4f  6417  modom  7098  finexdc  7197  ssfirab  7234  uzind4s  9969  zsupcllemstep  10640  bezoutlemmain  12753  nnwosdc  12794  cbvrald  16730  bj-bdfindes  16889  bj-findes  16921