Theorem nfeq 2340

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 2183	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2326	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2326	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1600	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1587	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1485	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   F/wnf 1471   e. wcel 2160   F/_wnfc 2319

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321

This theorem is referenced by:  nfel  2341  nfeq1  2342  nfeq2  2344  nfne  2453  raleqf  2682  rexeqf  2683  reueq1f  2684  rmoeq1f  2685  rabeqf  2742  sbceqg  3088  csbhypf  3110  nfiotadw  5196  nffn  5327  nffo  5452  fvmptdf  5619  mpteqb  5622  fvmptf  5624  eqfnfv2f  5633  dff13f  5787  ovmpos  6015  ov2gf  6016  ovmpodxf  6017  ovmpodf  6023  eqerlem  6584  sumeq2  11385  fsumadd  11432  prodeq1f  11578  prodeq2  11583  txcnp  14155  cnmpt11  14167  cnmpt21  14175  cnmptcom  14182  lgseisenlem2  14835