Theorem drnfc2 2357

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 2266	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf2 1748	. . 3 |- ( A. x x = y -> ( F/ z w e. A <-> F/ z w e. B ) )
4	3	dral2 1745	. 2 |- ( A. x x = y -> ( A. w F/ z w e. A <-> A. w F/ z w e. B ) )
5		df-nfc 2328	. 2 |- ( F/_ z A <-> A. w F/ z w e. A )
6		df-nfc 2328	. 2 |- ( F/_ z B <-> A. w F/ z w e. B )
7	4, 5, 6	3bitr4g 223	1 |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   F/wnf 1474   e. wcel 2167   F/_wnfc 2326

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-nfc 2328

This theorem is referenced by: (None)