Theorem bj-eqs 32663

Description: A lemma for substitutions, proved from Tarski's FOL. The version without DV( x, y) is true but requires ax-13 2246. The DV condition DV( x, ph) is necessary for both directions: consider substituting x = z for ph. (Contributed by BJ, 25-May-2021.)

Assertion

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

Distinct variable groups:   x, y   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem bj-eqs

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- ( ph -> ( x = y -> ph ) )
2	1	alrimiv 1855	. 2 |- ( ph -> A. x ( x = y -> ph ) )
3		exim 1761	. . 3 |- ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ph ) )
4		ax6ev 1890	. . . 4 |- E. x x = y
5		pm2.27 42	. . . 4 |- ( E. x x = y -> ( ( E. x x = y -> E. x ph ) -> E. x ph ) )
6	4, 5	ax-mp 5	. . 3 |- ( ( E. x x = y -> E. x ph ) -> E. x ph )
7		ax5e 1841	. . 3 |- ( E. x ph -> ph )
8	3, 6, 7	3syl 18	. 2 |- ( A. x ( x = y -> ph ) -> ph )
9	2, 8	impbii 199	1 |- ( ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

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

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-sb  32677