Theorem nfeq 2344

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 2187	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2330	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2330	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1600	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1587	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1485	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1362   = wceq 1364   F/wnf 1471   e. wcel 2164   F/_wnfc 2323

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325

This theorem is referenced by:  nfel  2345  nfeq1  2346  nfeq2  2348  nfne  2457  raleqf  2686  rexeqf  2687  reueq1f  2688  rmoeq1f  2689  rabeqf  2750  sbceqg  3096  csbhypf  3119  nfiotadw  5218  nffn  5350  nffo  5475  fvmptdf  5645  mpteqb  5648  fvmptf  5650  eqfnfv2f  5659  dff13f  5813  ovmpos  6042  ov2gf  6043  ovmpodxf  6044  ovmpodf  6050  eqerlem  6618  sumeq2  11502  fsumadd  11549  prodeq1f  11695  prodeq2  11700  txcnp  14439  cnmpt11  14451  cnmpt21  14459  cnmptcom  14466  lgseisenlem2  15187