Theorem ax11b 1707

Description: A bidirectional version of ax-11o 1704. (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 1703	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	imp 115	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( ph -> A. x ( x = y -> ph ) ) )
3		ax-4 1400	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
4	3	com12 27	. . 3 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
5	4	adantl 262	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( A. x ( x = y -> ph ) -> ph ) )
6	2, 5	impbid 120	1 |- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)