Theorem drsb1 1212

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).

Assertion

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

Proof of Theorem drsb1

Step	Hyp	Ref	Expression
1		equequ1 1171	. . . . 5 |- (x = y -> (x = z <-> y = z))
2	1	a4s 1020	. . . 4 |- (A.x x = y -> (x = z <-> y = z))
3	2	imbi1d 616	. . 3 |- (A.x x = y -> ((x = z -> ph) <-> (y = z -> ph)))
4	2	anbi1d 620	. . . 4 |- (A.x x = y -> ((x = z /\ ph) <-> (y = z /\ ph)))
5	4	drex1 1193	. . 3 |- (A.x x = y -> (E.x(x = z /\ ph) <-> E.y(y = z /\ ph)))
6	3, 5	anbi12d 631	. 2 |- (A.x x = y -> (((x = z -> ph) /\ E.x(x = z /\ ph)) <-> ((y = z -> ph) /\ E.y(y = z /\ ph))))
7		df-sb 1209	. 2 |- ([z / x]ph <-> ((x = z -> ph) /\ E.x(x = z /\ ph)))
8		df-sb 1209	. 2 |- ([z / y]ph <-> ((y = z -> ph) /\ E.y(y = z /\ ph)))
9	6, 7, 8	3bitr4g 558	1 |- (A.x x = y -> ([z / x]ph <-> [z / y]ph))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  [wsbc 1207

This theorem is referenced by:  sbequi 1265  sbco3 1295  sbcom 1296  sb9i 1301

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209

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