Theorem nfeq 2316

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 2159	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2302	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2302	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1577	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1564	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1462	1 |- F/ x A = B

Syntax hints:   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfel  2317  nfeq1  2318  nfeq2  2320  nfne  2429  raleqf  2657  rexeqf  2658  reueq1f  2659  rmoeq1f  2660  rabeqf  2716  sbceqg  3061  csbhypf  3083  nfiotadw  5156  nffn  5284  nffo  5409  fvmptdf  5573  mpteqb  5576  fvmptf  5578  eqfnfv2f  5587  dff13f  5738  ovmpos  5965  ov2gf  5966  ovmpodxf  5967  ovmpodf  5973  eqerlem  6532  sumeq2  11300  fsumadd  11347  prodeq1f  11493  prodeq2  11498  txcnp  12911  cnmpt11  12923  cnmpt21  12931  cnmptcom  12938