Theorem equs5 1783

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 1682	. 2 |- ( -. A. x x = y -> A. x -. A. x x = y )
2		hba1 1503	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
3		ax11o 1776	. . 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 1558	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 1312   E.wex 1451

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719

This theorem is referenced by:  sb3  1785  sb4  1786