Theorem drnfc1 2505

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/_xA <-> F/_yB))

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 2420	. . . 4 |- (A.x x = y -> (w e. A <-> w e. B))
3	2	drnf1 1969	. . 3 |- (A.x x = y -> ( F/x w e. A <-> F/y w e. B))
4	3	dral2 1966	. 2 |- (A.x x = y -> (A.w F/x w e. A <-> A.w F/y w e. B))
5		df-nfc 2478	. 2 |- ( F/_xA <-> A.w F/x w e. A)
6		df-nfc 2478	. 2 |- ( F/_yB <-> A.w F/y w e. B)
7	4, 5, 6	3bitr4g 279	1 |- (A.x x = y -> ( F/_xA <-> F/_yB))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfabd2  2507