Theorem bnj1541 30926

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1541.1	|- ( ph <-> ( ps /\ A =/= B ) )
bnj1541.2	|- -. ph

Assertion

Ref	Expression
bnj1541	|- ( ps -> A = B )

Proof of Theorem bnj1541

Step	Hyp	Ref	Expression
1		bnj1541.2	. . . 4 |- -. ph
2		bnj1541.1	. . . 4 |- ( ph <-> ( ps /\ A =/= B ) )
3	1, 2	mtbi 312	. . 3 |- -. ( ps /\ A =/= B )
4	3	imnani 439	. 2 |- ( ps -> -. A =/= B )
5		nne 2798	. 2 |- ( -. A =/= B <-> A = B )
6	4, 5	sylib 208	1 |- ( ps -> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-ne 2795

This theorem is referenced by:  bnj1312  31126  bnj1523  31139