Theorem drnfc1 2334

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 2245	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf1 1731	. . 3 |- ( A. x x = y -> ( F/ x w e. A <-> F/ y w e. B ) )
4	3	dral2 1729	. 2 |- ( A. x x = y -> ( A. w F/ x w e. A <-> A. w F/ y w e. B ) )
5		df-nfc 2306	. 2 |- ( F/_ x A <-> A. w F/ x w e. A )
6		df-nfc 2306	. 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 1351   = wceq 1353   F/wnf 1458   e. wcel 2146   F/_wnfc 2304

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-cleq 2168  df-clel 2171  df-nfc 2306

This theorem is referenced by: (None)