Theorem bj-dfssb2 32640

Description: An alternate definition of df-ssb 32620. Note that the use of a dummy variable in the definition df-ssb 32620 allows to use bj-sb56 32639 instead of equs45f 2350 and hence to avoid dependency on ax-13 2246 and to use ax-12 2047 only through bj-ax12 32634. Compare dfsb7 2455. (Contributed by BJ, 25-Dec-2020.)

Assertion

Ref	Expression
bj-dfssb2	|- ([ t/ x]b ph <-> E. y ( y = t /\ E. x ( x = y /\ ph ) ) )

Distinct variable groups:   y, t   x, y   ph, y

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-dfssb2

Step	Hyp	Ref	Expression
1		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
2		bj-sb56 32639	. 2 |- ( E. y ( y = t /\ A. x ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
3		bj-sb56 32639	. . . . 5 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
4	3	bicomi 214	. . . 4 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
5	4	anbi2i 730	. . 3 |- ( ( y = t /\ A. x ( x = y -> ph ) ) <-> ( y = t /\ E. x ( x = y /\ ph ) ) )
6	5	exbii 1774	. 2 |- ( E. y ( y = t /\ A. x ( x = y -> ph ) ) <-> E. y ( y = t /\ E. x ( x = y /\ ph ) ) )
7	1, 2, 6	3bitr2i 288	1 |- ([ t/ x]b ph <-> E. y ( y = t /\ E. x ( x = y /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704  [wssb 32619

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by:  bj-ssbn  32641