Theorem drsb1 1680

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 1598	. . . . 5 |- ( x = y -> ( x = z <-> y = z ) )
2	1	sps 1430	. . . 4 |- ( A. x x = y -> ( x = z <-> y = z ) )
3	2	imbi1d 220	. . 3 |- ( A. x x = y -> ( ( x = z -> ph ) <-> ( y = z -> ph ) ) )
4	2	anbi1d 438	. . . 4 |- ( A. x x = y -> ( ( x = z /\ ph ) <-> ( y = z /\ ph ) ) )
5	4	drex1 1679	. . 3 |- ( A. x x = y -> ( E. x ( x = z /\ ph ) <-> E. y ( y = z /\ ph ) ) )
6	3, 5	anbi12d 442	. 2 |- ( A. x x = y -> ( ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) ) )
7		df-sb 1646	. 2 |- ( [ z / x ] ph <-> ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) )
8		df-sb 1646	. 2 |- ( [ z / y ] ph <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) )
9	6, 7, 8	3bitr4g 212	1 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbequi  1720  nfsbxy  1818  nfsbxyt  1819  sbcomxyyz  1846  iotaeq  4875