Theorem sbequ 1768

Description: An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbequ

Step	Hyp	Ref	Expression
1		sbequi 1767	. 2 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )
2		sbequi 1767	. . 3 |- ( y = x -> ( [ y / z ] ph -> [ x / z ] ph ) )
3	2	equcoms 1641	. 2 |- ( x = y -> ( [ y / z ] ph -> [ x / z ] ph ) )
4	1, 3	impbid 127	1 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )

Syntax hints:   -> wi 4   <-> wb 103   [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  drsb2  1769  sbco2vlem  1868  sbco2yz  1885  sbcocom  1892  sb10f  1919  hbsb4  1936  nfsb4or  1947  sb8eu  1961  sb8euh  1971  cbvab  2210  cbvralf  2584  cbvrexf  2585  cbvreu  2588  cbvralsv  2601  cbvrexsv  2602  cbvrab  2617  cbvreucsf  2992  cbvrabcsf  2993  sbss  3390  disjiun  3840  cbvopab1  3911  cbvmpt  3933  tfis  4398  findes  4418  cbviota  4985  sb8iota  4987  cbvriota  5618  uzind4s  9076  bezoutlemmain  11261  cbvrald  11643  setindft  11815