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  2669  rexeqf  2670  reueq1f  2671  rmoeq1f  2672  rabeqf  2728  sbceqg  3074  csbhypf  3096  nfiotadw  5182  nffn  5313  nffo  5438  fvmptdf  5604  mpteqb  5607  fvmptf  5609  eqfnfv2f  5618  dff13f  5771  ovmpos  5998  ov2gf  5999  ovmpodxf  6000  ovmpodf  6006  eqerlem  6566  sumeq2  11367  fsumadd  11414  prodeq1f  11560  prodeq2  11565  txcnp  13774  cnmpt11  13786  cnmpt21  13794  cnmptcom  13801  lgseisenlem2  14454