Theorem nfeq 2237

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfnfc.1	|- F/_ x A
nfeq.2	|- F/_ x B

Assertion

Ref	Expression
nfeq	|- F/ x A = B

Proof of Theorem nfeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2083	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2223	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2223	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1527	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1514	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1409	1 |- F/ x A = B

Syntax hints:   <-> wb 104   A.wal 1288   = wceq 1290   F/wnf 1395   e. wcel 1439   F/_wnfc 2216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218

This theorem is referenced by:  nfel  2238  nfeq1  2239  nfeq2  2241  nfne  2349  raleqf  2559  rexeqf  2560  reueq1f  2561  rmoeq1f  2562  rabeqf  2610  sbceqg  2948  csbhypf  2967  nfiotadxy  4996  nffn  5123  nffo  5245  fvmptdf  5403  mpteqb  5406  fvmptf  5408  eqfnfv2f  5415  dff13f  5563  ovmpt2s  5782  ov2gf  5783  ovmpt2dxf  5784  ovmpt2df  5790  eqerlem  6337  sumeq2  10809  fsumadd  10861