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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   [wsb 1815

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

This theorem is used 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  4205  cbvopab1s  4207  csbopabg  4210  cbvmptf  4227  cbvmpt  4228  opelopabsb  4404  frind  4499  tfis  4732  findes  4752  opeliunxp  4832  ralxpf  4928  rexxpf  4929  cbviota  5345  csbiotag  5373  cbvriota  6053  csbriotag  6055  abrexex2g  6353  opabex3d  6354  opabex3  6355  abrexex2  6357  dfoprab4f  6431  modom  7112  finexdc  7211  ssfirab  7248  uzind4s  10006  zsupcllemstep  10679  bezoutlemmain  12800  nnwosdc  12841  cbvrald  17021  bj-bdfindes  17180  bj-findes  17212