Theorem ax11b 1819

Description: A bidirectional version of ax-11o 1816. (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 1815	. . 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 1503	. . . 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 1346

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by: (None)