Theorem nfeqd 2503

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	|- (ph -> F/_xA)
nfeqd.2	|- (ph -> F/_xB)

Assertion

Ref	Expression
nfeqd	|- (ph -> F/x A = B)

Proof of Theorem nfeqd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2347	. 2 |- (A = B <-> A.y(y e. A <-> y e. B))
2		nfv 1619	. . 3 |- F/yph
3		nfeqd.1	. . . . 5 |- (ph -> F/_xA)
4	3	nfcrd 2502	. . . 4 |- (ph -> F/x y e. A)
5		nfeqd.2	. . . . 5 |- (ph -> F/_xB)
6	5	nfcrd 2502	. . . 4 |- (ph -> F/x y e. B)
7	4, 6	nfbid 1832	. . 3 |- (ph -> F/x(y e. A <-> y e. B))
8	2, 7	nfald 1852	. 2 |- (ph -> F/xA.y(y e. A <-> y e. B))
9	1, 8	nfxfrd 1571	1 |- (ph -> F/x A = B)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-nfc 2478

This theorem is referenced by:  nfeld  2504  nfned  2612  vtoclgft  2905  sbcralt  3118  csbiebt  3172  dfnfc2  3909  nfiotad  4342  iota2df  4365  dfid3  4768  oprabid  5550