Theorem axc16gb 2136

Description: Biconditional strengthening of axc16g 2134. (Contributed by NM, 15-May-1993.)

Assertion

Ref	Expression
axc16gb	|- ( A. x x = y -> ( ph <-> A. z ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem axc16gb

Step	Hyp	Ref	Expression
1		axc16g 2134	. 2 |- ( A. x x = y -> ( ph -> A. z ph ) )
2		sp 2053	. 2 |- ( A. z ph -> ph )
3	1, 2	impbid1 215	1 |- ( A. x x = y -> ( ph <-> A. z ph ) )

Syntax hints:   -> wi 4   <-> wb 196   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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  sbal  2462