Theorem drnf2 1787

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 1784	. . . 4 |- ( A. x x = y -> ( A. z ph <-> A. z ps ) )
3	1, 2	imbi12d 234	. . 3 |- ( A. x x = y -> ( ( ph -> A. z ph ) <-> ( ps -> A. z ps ) ) )
4	3	dral2 1784	. 2 |- ( A. x x = y -> ( A. z ( ph -> A. z ph ) <-> A. z ( ps -> A. z ps ) ) )
5		df-nf 1514	. 2 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6		df-nf 1514	. 2 |- ( F/ z ps <-> A. z ( ps -> A. z ps ) )
7	4, 5, 6	3bitr4g 223	1 |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   F/wnf 1513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  nfsbxy  2002  nfsbxyt  2003  drnfc2  2410