Theorem sbequ 1864

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 1863	. 2 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )
2		sbequi 1863	. . 3 |- ( y = x -> ( [ y / z ] ph -> [ x / z ] ph ) )
3	2	equcoms 1732	. 2 |- ( x = y -> ( [ y / z ] ph -> [ x / z ] ph ) )
4	1, 3	impbid 129	1 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )

Syntax hints:   -> wi 4   <-> wb 105   [wsb 1786

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  drsb2  1865  sbco2vlem  1973  sbco2v  1977  sbco2yz  1992  sbcocom  1999  sb10f  2024  hbsb4  2041  nfsb4or  2050  sb8eu  2068  sb8euh  2078  cbvab  2331  cbvralf  2733  cbvrexf  2734  cbvreu  2740  cbvralsv  2758  cbvrexsv  2759  cbvrab  2774  cbvreucsf  3166  cbvrabcsf  3167  sbss  3576  disjiun  4054  cbvopab1  4133  cbvmpt  4155  tfis  4649  findes  4669  cbviota  5256  sb8iota  5258  cbvriota  5933  uzind4s  9746  bezoutlemmain  12434  cbvrald  15924  setindft  16100