Theorem drnfc1 2366

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 2276	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf1 1757	. . 3 |- ( A. x x = y -> ( F/ x w e. A <-> F/ y w e. B ) )
4	3	dral2 1755	. 2 |- ( A. x x = y -> ( A. w F/ x w e. A <-> A. w F/ y w e. B ) )
5		df-nfc 2338	. 2 |- ( F/_ x A <-> A. w F/ x w e. A )
6		df-nfc 2338	. 2 |- ( F/_ y B <-> A. w F/ y w e. B )
7	4, 5, 6	3bitr4g 223	1 |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   e. wcel 2177   F/_wnfc 2336

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2199  df-clel 2202  df-nfc 2338

This theorem is referenced by: (None)