Theorem nfeq 2327

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 2171	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2313	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2313	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1589	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1576	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1474	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1460   e. wcel 2148   F/_wnfc 2306

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  nfel  2328  nfeq1  2329  nfeq2  2331  nfne  2440  raleqf  2668  rexeqf  2669  reueq1f  2670  rmoeq1f  2671  rabeqf  2727  sbceqg  3073  csbhypf  3095  nfiotadw  5181  nffn  5312  nffo  5437  fvmptdf  5603  mpteqb  5606  fvmptf  5608  eqfnfv2f  5617  dff13f  5770  ovmpos  5997  ov2gf  5998  ovmpodxf  5999  ovmpodf  6005  eqerlem  6565  sumeq2  11366  fsumadd  11413  prodeq1f  11559  prodeq2  11564  txcnp  13741  cnmpt11  13753  cnmpt21  13761  cnmptcom  13768  lgseisenlem2  14421