Theorem drnf2 1622

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

Hypothesis

Ref	Expression
drex2.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
drnf2	|- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )

Proof of Theorem drnf2

Step	Hyp	Ref	Expression
1		drex2.1	. . . 4 |- ( A. x x = y -> ( ph <-> ps ) )
2	1	dral2 1619	. . . 4 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
3	1, 2	imbi12d 223	. . 3 |- ( A. x x = y -> ( ( ph -> A. z ph ) <-> ( ps -> A. z ps ) ) )
4	3	dral2 1619	. 2 |- ( A. x x = y -> ( A. z ( ph -> A. z ph ) <-> A. z ( ps -> A. z ps ) ) )
5		df-nf 1350	. 2 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6		df-nf 1350	. 2 |- ( F/ z ps <-> A. z ( ps -> A. z ps ) )
7	4, 5, 6	3bitr4g 212	1 |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  nfsbxy  1818  nfsbxyt  1819  drnfc2  2195