Theorem ax12b 2345

Description: A bidirectional version of axc15 2303. (Contributed by NM, 30-Jun-2006.)

Assertion

Ref	Expression
ax12b	|- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Proof of Theorem ax12b

Step	Hyp	Ref	Expression
1		axc15 2303	. . 3 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
2	1	imp 445	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( ph -> A. x ( x = y -> ph ) ) )
3		sp 2053	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
4	3	com12 32	. . 3 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
5	4	adantl 482	. 2 |- ( ( -. A. x x = y /\ x = y ) -> ( A. x ( x = y -> ph ) -> ph ) )
6	2, 5	impbid 202	1 |- ( ( -. A. x x = y /\ x = y ) -> ( ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)