Theorem dral2 2173

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 2174. (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 2165	. 2 |- F/ z A. x x = y
2		dral1.1	. 2 |- ( A. x x = y -> ( ph <-> ps ) )
3	1, 2	albid 1983	1 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676

This theorem is referenced by:  dral1ALT  2175  sbal1  2309  sbal2  2310  drnfc1  2629  drnfc2  2630  axpownd  9044  wl-sbalnae  31962