Theorem ax11b 1826

Description: A bidirectional version of ax-11o 1823. (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 1822	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	imp 124	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( ph -> A. x ( x = y -> ph ) ) )
3		ax-4 1510	. . . 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 277	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( A. x ( x = y -> ph ) -> ph ) )
6	2, 5	impbid 129	1 |- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by: (None)