Theorem drsb1 1721

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 1639	. . . . 5 |- ( x = y -> ( x = z <-> y = z ) )
2	1	sps 1471	. . . 4 |- ( A. x x = y -> ( x = z <-> y = z ) )
3	2	imbi1d 229	. . 3 |- ( A. x x = y -> ( ( x = z -> ph ) <-> ( y = z -> ph ) ) )
4	2	anbi1d 453	. . . 4 |- ( A. x x = y -> ( ( x = z /\ ph ) <-> ( y = z /\ ph ) ) )
5	4	drex1 1720	. . 3 |- ( A. x x = y -> ( E. x ( x = z /\ ph ) <-> E. y ( y = z /\ ph ) ) )
6	3, 5	anbi12d 457	. 2 |- ( A. x x = y -> ( ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) ) )
7		df-sb 1687	. 2 |- ( [ z / x ] ph <-> ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) )
8		df-sb 1687	. 2 |- ( [ z / y ] ph <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) )
9	6, 7, 8	3bitr4g 221	1 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-sb 1687

This theorem is referenced by:  sbequi  1761  nfsbxy  1860  nfsbxyt  1861  sbcomxyyz  1888  iotaeq  4905