Theorem nfeq 2356

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 2199	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2342	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2342	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1612	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1599	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1497	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1483   e. wcel 2176   F/_wnfc 2335

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  nfel  2357  nfeq1  2358  nfeq2  2360  nfne  2469  raleqf  2698  rexeqf  2699  reueq1f  2700  rmoeq1f  2701  rabeqf  2762  sbceqg  3109  csbhypf  3132  nfiotadw  5236  nffn  5371  nffo  5499  fvmptdf  5669  mpteqb  5672  fvmptf  5674  eqfnfv2f  5683  dff13f  5841  ovmpos  6071  ov2gf  6072  ovmpodxf  6073  ovmpodf  6079  eqerlem  6653  sumeq2  11703  fsumadd  11750  prodeq1f  11896  prodeq2  11901  txcnp  14776  cnmpt11  14788  cnmpt21  14796  cnmptcom  14803  dvmptfsum  15230  lgseisenlem2  15581