Theorem bj-ssbid2ALT 32646

Description: Alternate proof of bj-ssbid2 32645, not using bj-ssbequ2 32643. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ssbid2ALT	|- ([ x/ x]b ph -> ph )

Proof of Theorem bj-ssbid2ALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssb 32620	. 2 |- ([ x/ x]b ph <-> A. y ( y = x -> A. x ( x = y -> ph ) ) )
2		sp 2053	. . . . 5 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
3	2	imim2i 16	. . . 4 |- ( ( y = x -> A. x ( x = y -> ph ) ) -> ( y = x -> ( x = y -> ph ) ) )
4	3	alimi 1739	. . 3 |- ( A. y ( y = x -> A. x ( x = y -> ph ) ) -> A. y ( y = x -> ( x = y -> ph ) ) )
5		pm2.21 120	. . . . . 6 |- ( -. y = x -> ( y = x -> ph ) )
6		equcomi 1944	. . . . . . 7 |- ( y = x -> x = y )
7	6	imim1i 63	. . . . . 6 |- ( ( x = y -> ph ) -> ( y = x -> ph ) )
8	5, 7	ja 173	. . . . 5 |- ( ( y = x -> ( x = y -> ph ) ) -> ( y = x -> ph ) )
9	8	alimi 1739	. . . 4 |- ( A. y ( y = x -> ( x = y -> ph ) ) -> A. y ( y = x -> ph ) )
10		ax6ev 1890	. . . 4 |- E. y y = x
11		19.23v 1902	. . . . 5 |- ( A. y ( y = x -> ph ) <-> ( E. y y = x -> ph ) )
12	11	biimpi 206	. . . 4 |- ( A. y ( y = x -> ph ) -> ( E. y y = x -> ph ) )
13	9, 10, 12	mpisyl 21	. . 3 |- ( A. y ( y = x -> ( x = y -> ph ) ) -> ph )
14	4, 13	syl 17	. 2 |- ( A. y ( y = x -> A. x ( x = y -> ph ) ) -> ph )
15	1, 14	sylbi 207	1 |- ([ x/ x]b ph -> ph )

Syntax hints:   -> wi 4   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-12 2047

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

This theorem is referenced by: (None)