Theorem equs5 2351

Description: Lemma used in proofs of substitution properties. If there is a dv condition on x , y, then sb56 2150 can be used instead; if y is not free in ph, then equs45f 2350 can be used. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.)

Assertion

Ref	Expression
equs5	|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )

Proof of Theorem equs5

Step	Hyp	Ref	Expression
1		nfna1 2029	. . 3 |- F/ x -. A. x x = y
2		nfa1 2028	. . 3 |- F/ x A. x ( x = y -> ph )
3		axc15 2303	. . . 4 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
4	3	impd 447	. . 3 |- ( -. A. x x = y -> ( ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
5	1, 2, 4	exlimd 2087	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
6		equs4 2290	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7	5, 6	impbid1 215	1 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   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  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  sb3  2355  sb4  2356  bj-sbsb  32824