Theorem drsb1 2022

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 1684	. . . . 5 |- (x = y -> (x = z <-> y = z))
2	1	sps 1754	. . . 4 |- (A.x x = y -> (x = z <-> y = z))
3	2	imbi1d 308	. . 3 |- (A.x x = y -> ((x = z -> ph) <-> (y = z -> ph)))
4	2	anbi1d 685	. . . 4 |- (A.x x = y -> ((x = z /\ ph) <-> (y = z /\ ph)))
5	4	drex1 1967	. . 3 |- (A.x x = y -> (E.x(x = z /\ ph) <-> E.y(y = z /\ ph)))
6	3, 5	anbi12d 691	. 2 |- (A.x x = y -> (((x = z -> ph) /\ E.x(x = z /\ ph)) <-> ((y = z -> ph) /\ E.y(y = z /\ ph))))
7		df-sb 1649	. 2 |- ([z / x]ph <-> ((x = z -> ph) /\ E.x(x = z /\ ph)))
8		df-sb 1649	. 2 |- ([z / y]ph <-> ((y = z -> ph) /\ E.y(y = z /\ ph)))
9	6, 7, 8	3bitr4g 279	1 |- (A.x x = y -> ([z / x]ph <-> [z / y]ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbequi  2059  nfsb4t  2080  sbco3  2088  sbcom  2089  sb9i  2094  iotaeq  4348