Theorem nfeq 2394

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 2228	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2380	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2380	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1638	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1625	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1523	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1396   = wceq 1398   F/wnf 1509   e. wcel 2205   F/_wnfc 2373

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-cleq 2227  df-clel 2230  df-nfc 2375

This theorem is referenced by:  nfel  2395  nfeq1  2396  nfeq2  2398  nfne  2507  raleqf  2739  rexeqf  2740  reueq1f  2741  rmoeq1f  2742  rabeqf  2805  sbceqg  3157  csbhypf  3180  nfiotadw  5320  nffn  5457  nffo  5594  fvmptdf  5770  mpteqb  5773  fvmptf  5775  eqfnfv2f  5784  dff13f  5949  ovmpos  6185  ov2gf  6186  ovmpodxf  6187  ovmpodf  6193  eqerlem  6811  sumeq2  12069  fsumadd  12117  prodeq1f  12263  prodeq2  12268  txcnp  15262  cnmpt11  15274  cnmpt21  15282  cnmptcom  15289  dvmptfsum  15716  lgseisenlem2  16070