Theorem nfeq 2358

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 2201	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2344	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2344	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1613	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1600	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1498	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   e. wcel 2178   F/_wnfc 2337

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339

This theorem is referenced by:  nfel  2359  nfeq1  2360  nfeq2  2362  nfne  2471  raleqf  2701  rexeqf  2702  reueq1f  2703  rmoeq1f  2704  rabeqf  2766  sbceqg  3117  csbhypf  3140  nfiotadw  5254  nffn  5389  nffo  5519  fvmptdf  5690  mpteqb  5693  fvmptf  5695  eqfnfv2f  5704  dff13f  5862  ovmpos  6092  ov2gf  6093  ovmpodxf  6094  ovmpodf  6100  eqerlem  6674  sumeq2  11785  fsumadd  11832  prodeq1f  11978  prodeq2  11983  txcnp  14858  cnmpt11  14870  cnmpt21  14878  cnmptcom  14885  dvmptfsum  15312  lgseisenlem2  15663