Theorem drnf2 1619

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/zph <-> F/zps))

Proof of Theorem drnf2

Step	Hyp	Ref	Expression
1		drex2.1	. . . 4 |- (A.x x = y -> (ph <-> ps))
2	1	dral2 1616	. . . 4 |- (A.x x = y -> (A.zph <-> A.zps))
3	1, 2	imbi12d 223	. . 3 |- (A.x x = y -> ((ph -> A.zph) <-> (ps -> A.zps)))
4	3	dral2 1616	. 2 |- (A.x x = y -> (A.z(ph -> A.zph) <-> A.z(ps -> A.zps)))
5		df-nf 1347	. 2 |- ( F/zph <-> A.z(ph -> A.zph))
6		df-nf 1347	. 2 |- ( F/zps <-> A.z(ps -> A.zps))
7	4, 5, 6	3bitr4g 212	1 |- (A.x x = y -> ( F/zph <-> F/zps))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   F/wnf 1346

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  nfsbxy  1815  nfsbxyt  1816  drnfc2  2192