Theorem drnfc2 2297

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
drnfc2	|- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) )

Proof of Theorem drnfc2

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 2207	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf2 1712	. . 3 |- ( A. x x = y -> ( F/ z w e. A <-> F/ z w e. B ) )
4	3	dral2 1709	. 2 |- ( A. x x = y -> ( A. w F/ z w e. A <-> A. w F/ z w e. B ) )
5		df-nfc 2268	. 2 |- ( F/_ z A <-> A. w F/ z w e. A )
6		df-nfc 2268	. 2 |- ( F/_ z B <-> A. w F/ z w e. B )
7	4, 5, 6	3bitr4g 222	1 |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2266

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2130  df-clel 2133  df-nfc 2268

This theorem is referenced by: (None)