Theorem dral2 2023

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.) Revised to allow a shortening of dral1 2024. (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 2013	. 2 |- F/ z A. x x = y
2		dral1.1	. 2 |- ( A. x x = y -> ( ph <-> ps ) )
3	1, 2	albid 1821	1 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591

This theorem is referenced by:  dral1ALT  2025  sbal1  2178  sbal1OLD  2179  sbal2  2180  drnfc1  2631  drnfc2  2632  axpownd  8871  wl-sbalnae  28529