Theorem drsb1 1772

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
drsb1	|- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )

Proof of Theorem drsb1

Step	Hyp	Ref	Expression
1		equequ1 1689	. . . . 5 |- ( x = y -> ( x = z <-> y = z ) )
2	1	sps 1518	. . . 4 |- ( A. x x = y -> ( x = z <-> y = z ) )
3	2	imbi1d 230	. . 3 |- ( A. x x = y -> ( ( x = z -> ph ) <-> ( y = z -> ph ) ) )
4	2	anbi1d 461	. . . 4 |- ( A. x x = y -> ( ( x = z /\ ph ) <-> ( y = z /\ ph ) ) )
5	4	drex1 1771	. . 3 |- ( A. x x = y -> ( E. x ( x = z /\ ph ) <-> E. y ( y = z /\ ph ) ) )
6	3, 5	anbi12d 465	. 2 |- ( A. x x = y -> ( ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) ) )
7		df-sb 1737	. 2 |- ( [ z / x ] ph <-> ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) )
8		df-sb 1737	. 2 |- ( [ z / y ] ph <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) )
9	6, 7, 8	3bitr4g 222	1 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   E.wex 1469   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-sb 1737

This theorem is referenced by:  sbequi  1812  nfsbxy  1916  nfsbxyt  1917  sbcomxyyz  1946  iotaeq  5104