Theorem nfeld 2408

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

Hypotheses

Ref	Expression
nfeqd.1	|- ( ph -> F/_ x A )
nfeqd.2	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfeld	|- ( ph -> F/ x A e. B )

Proof of Theorem nfeld

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2234	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1581	. . 3 |- F/ y ph
3		nfcvd 2393	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2407	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2406	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1621	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1814	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1528	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   F/wnf 1513   E.wex 1545   e. wcel 2209   F/_wnfc 2379

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-nfc 2381

This theorem is referenced by:  nfneld  2523  nfraldw  2582  nfraldxy  2583  nfrexdxy  2584  nfreudxy  2725  nfsbc1d  3068  nfsbcd  3071  sbcrext  3129  nfsbcdw  3181  nfbrd  4171  nfriotadxy  6037  nfixpxy  6989