Theorem drnfc1 2325

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypothesis

Ref	Expression
drnfc1.1	|- ( A. x x = y -> A = B )

Assertion

Ref	Expression
drnfc1	|- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) )

Proof of Theorem drnfc1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drnfc1.1	. . . . 5 |- ( A. x x = y -> A = B )
2	1	eleq2d 2236	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf1 1721	. . 3 |- ( A. x x = y -> ( F/ x w e. A <-> F/ y w e. B ) )
4	3	dral2 1719	. 2 |- ( A. x x = y -> ( A. w F/ x w e. A <-> A. w F/ y w e. B ) )
5		df-nfc 2297	. 2 |- ( F/_ x A <-> A. w F/ x w e. A )
6		df-nfc 2297	. 2 |- ( F/_ y B <-> A. w F/ y w e. B )
7	4, 5, 6	3bitr4g 222	1 |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   F/_wnfc 2295

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-ie1 1481  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  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by: (None)