Theorem nfeq 2380

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 2223	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2366	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2366	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1635	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1622	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1520	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by:  nfel  2381  nfeq1  2382  nfeq2  2384  nfne  2493  raleqf  2724  rexeqf  2725  reueq1f  2726  rmoeq1f  2727  rabeqf  2789  sbceqg  3140  csbhypf  3163  nfiotadw  5281  nffn  5417  nffo  5547  fvmptdf  5722  mpteqb  5725  fvmptf  5727  eqfnfv2f  5736  dff13f  5894  ovmpos  6128  ov2gf  6129  ovmpodxf  6130  ovmpodf  6136  eqerlem  6711  sumeq2  11870  fsumadd  11917  prodeq1f  12063  prodeq2  12068  txcnp  14945  cnmpt11  14957  cnmpt21  14965  cnmptcom  14972  dvmptfsum  15399  lgseisenlem2  15750