Theorem drnf1 1710

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

Proof of Theorem drnf1

Step	Hyp	Ref	Expression
1		drex2.1	. . . 4 |- ( A. x x = y -> ( ph <-> ps ) )
2	1	dral1 1707	. . . 4 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
3	1, 2	imbi12d 233	. . 3 |- ( A. x x = y -> ( ( ph -> A. x ph ) <-> ( ps -> A. y ps ) ) )
4	3	dral1 1707	. 2 |- ( A. x x = y -> ( A. x ( ph -> A. x ph ) <-> A. y ( ps -> A. y ps ) ) )
5		df-nf 1438	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1438	. 2 |- ( F/ y ps <-> A. y ( ps -> A. y ps ) )
7	4, 5, 6	3bitr4g 222	1 |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   F/wnf 1437

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 1424  ax-7 1425  ax-gen 1426  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511

This theorem depends on definitions:  df-bi 116  df-nf 1438

This theorem is referenced by:  drnfc1  2313