Theorem bj-sb 32677

Description: A weak variant of sbid2 2413 not requiring ax-13 2246 nor ax-10 2019. On top of Tarski's FOL, one implication requires only ax12v 2048, and the other requires only sp 2053. (Contributed by BJ, 25-May-2021.)

Assertion

Ref	Expression
bj-sb	|- ( ph <-> A. y ( y = x -> A. x ( x = y -> ph ) ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem bj-sb

Step	Hyp	Ref	Expression
1		ax12v 2048	. . . . 5 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
2	1	equcoms 1947	. . . 4 |- ( y = x -> ( ph -> A. x ( x = y -> ph ) ) )
3	2	com12 32	. . 3 |- ( ph -> ( y = x -> A. x ( x = y -> ph ) ) )
4	3	alrimiv 1855	. 2 |- ( ph -> A. y ( y = x -> A. x ( x = y -> ph ) ) )
5		sp 2053	. . . . . . 7 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
6	5	com12 32	. . . . . 6 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
7	6	equcoms 1947	. . . . 5 |- ( y = x -> ( A. x ( x = y -> ph ) -> ph ) )
8	7	a2i 14	. . . 4 |- ( ( y = x -> A. x ( x = y -> ph ) ) -> ( y = x -> ph ) )
9	8	alimi 1739	. . 3 |- ( A. y ( y = x -> A. x ( x = y -> ph ) ) -> A. y ( y = x -> ph ) )
10		bj-eqs 32663	. . 3 |- ( ph <-> A. y ( y = x -> ph ) )
11	9, 10	sylibr 224	. 2 |- ( A. y ( y = x -> A. x ( x = y -> ph ) ) -> ph )
12	4, 11	impbii 199	1 |- ( ph <-> A. y ( y = x -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481

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

This theorem is referenced by: (None)