Theorem equs5 1817

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

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		hbnae 1709	. 2 |- ( -. A. x x = y -> A. x -. A. x x = y )
2		hba1 1528	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
3		ax11o 1810	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
4	3	impd 252	. 2 |- ( -. A. x x = y -> ( ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
5	1, 2, 4	exlimdh 1584	1 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751

This theorem is referenced by:  sb3  1819  sb4  1820