Theorem ax11b 1803

Description: A bidirectional version of ax-11o 1800. (Contributed by NM, 30-Jun-2006.)

Assertion

Ref	Expression
ax11b	|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Proof of Theorem ax11b

Step	Hyp	Ref	Expression
1		ax11o 1799	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	imp 123	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( ph -> A. x ( x = y -> ph ) ) )
3		ax-4 1487	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
4	3	com12 30	. . 3 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
5	4	adantl 275	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( A. x ( x = y -> ph ) -> ph ) )
6	2, 5	impbid 128	1 |- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330

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-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740

This theorem is referenced by: (None)