Theorem dral1 2325

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, 24-Nov-1994.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 6-Sep-2018.)

Hypothesis

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

Assertion

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

Proof of Theorem dral1

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ x A. x x = y
2		dral1.1	. . 3 |- ( A. x x = y -> ( ph <-> ps ) )
3	1, 2	albid 2090	. 2 |- ( A. x x = y -> ( A. x ph <-> A. x ps ) )
4		axc11 2314	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
5		axc11r 2187	. . 3 |- ( A. x x = y -> ( A. y ps -> A. x ps ) )
6	4, 5	impbid 202	. 2 |- ( A. x x = y -> ( A. x ps <-> A. y ps ) )
7	3, 6	bitrd 268	1 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  drex1  2327  drnf1  2329  ax12OLD  2341  axc16gALT  2367  sb9  2426  ralcom2  3104  axpownd  9423  wl-dral1d  33318  wl-ax11-lem5  33366  wl-ax11-lem8  33369  wl-ax11-lem9  33370