Theorem drnfc2 2783

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 2687	. . . 4 |- ( A. x x = y -> ( w e. A <-> w e. B ) )
3	2	drnf2 2330	. . 3 |- ( A. x x = y -> ( F/ z w e. A <-> F/ z w e. B ) )
4	3	dral2 2324	. 2 |- ( A. x x = y -> ( A. w F/ z w e. A <-> A. w F/ z w e. B ) )
5		df-nfc 2753	. 2 |- ( F/_ z A <-> A. w F/ z w e. A )
6		df-nfc 2753	. 2 |- ( F/_ z B <-> A. w F/ z w e. B )
7	4, 5, 6	3bitr4g 303	1 |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)