Theorem drnf2 1722

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 1719	. . . 4 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
3	1, 2	imbi12d 233	. . 3 |- ( A. x x = y -> ( ( ph -> A. z ph ) <-> ( ps -> A. z ps ) ) )
4	3	dral2 1719	. 2 |- ( A. x x = y -> ( A. z ( ph -> A. z ph ) <-> A. z ( ps -> A. z ps ) ) )
5		df-nf 1449	. 2 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6		df-nf 1449	. 2 |- ( F/ z ps <-> A. z ( ps -> A. z ps ) )
7	4, 5, 6	3bitr4g 222	1 |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   F/wnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  nfsbxy  1930  nfsbxyt  1931  drnfc2  2326