Theorem drsb1 2377

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, 2-Jun-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 1952	. . . . 5 |- ( x = y -> ( x = z <-> y = z ) )
2	1	sps 2055	. . . 4 |- ( A. x x = y -> ( x = z <-> y = z ) )
3	2	imbi1d 331	. . 3 |- ( A. x x = y -> ( ( x = z -> ph ) <-> ( y = z -> ph ) ) )
4	2	anbi1d 741	. . . 4 |- ( A. x x = y -> ( ( x = z /\ ph ) <-> ( y = z /\ ph ) ) )
5	4	drex1 2327	. . 3 |- ( A. x x = y -> ( E. x ( x = z /\ ph ) <-> E. y ( y = z /\ ph ) ) )
6	3, 5	anbi12d 747	. 2 |- ( A. x x = y -> ( ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) ) )
7		df-sb 1881	. 2 |- ( [ z / x ] ph <-> ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) )
8		df-sb 1881	. 2 |- ( [ z / y ] ph <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) )
9	6, 7, 8	3bitr4g 303	1 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco3  2417  iotaeq  5859