Theorem drnf1 2021

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypothesis

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

Assertion

Ref	Expression
drnf1	|- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )

Proof of Theorem drnf1

Step	Hyp	Ref	Expression
1		dral1.1	. . . 4 |- ( A. x x = y -> ( ph <-> ps ) )
2	1	dral1 2017	. . . 4 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
3	1, 2	imbi12d 320	. . 3 |- ( A. x x = y -> ( ( ph -> A. x ph ) <-> ( ps -> A. y ps ) ) )
4	3	dral1 2017	. 2 |- ( A. x x = y -> ( A. x ( ph -> A. x ph ) <-> A. y ( ps -> A. y ps ) ) )
5		df-nf 1590	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1590	. 2 |- ( F/ y ps <-> A. y ( ps -> A. y ps ) )
7	4, 5, 6	3bitr4g 288	1 |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1367   F/wnf 1589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792  ax-13 1943

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590

This theorem is referenced by:  nfald2  2023  drnfc1  2607  wl-nfs1t  28393