Theorem dral2 2125

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2126. (Revised by Wolf Lammen, 4-Mar-2018.)

Hypothesis

Ref	Expression
dral1.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
dral2	|- ( A. x x = y -> ( A. z ph <-> A. z ps ) )

Proof of Theorem dral2

Step	Hyp	Ref	Expression
1		nfae 2115	. 2 |- F/ z A. x x = y
2		dral1.1	. 2 |- ( A. x x = y -> ( ph <-> ps ) )
3	1, 2	albid 1940	1 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662

This theorem is referenced by:  dral1ALT  2127  sbal1  2259  sbal2  2260  drnfc1  2599  drnfc2  2600  axpownd  9033  wl-sbalnae  31856