Theorem bj-ssbid1ALT 31325

Description: Alternate proof of bj-ssbid1 31324, not using bj-ssbequ1 31321. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ssbid1ALT	|- ( ph -> [ x/ x]b ph )

Proof of Theorem bj-ssbid1ALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax12v 1951	. . . . 5 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
2	1	equcoms 1872	. . . 4 |- ( y = x -> ( ph -> A. x ( x = y -> ph ) ) )
3	2	com12 31	. . 3 |- ( ph -> ( y = x -> A. x ( x = y -> ph ) ) )
4	3	alrimiv 1781	. 2 |- ( ph -> A. y ( y = x -> A. x ( x = y -> ph ) ) )
5		df-ssb 31297	. 2 |- ([ x/ x]b ph <-> A. y ( y = x -> A. x ( x = y -> ph ) ) )
6	4, 5	sylibr 217	1 |- ( ph -> [ x/ x]b ph )

Syntax hints:   -> wi 4   A.wal 1450  [wssb 31296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-ssb 31297

This theorem is referenced by: (None)