Theorem nfeld 2335

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 2173	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1528	. . 3 |- F/ y ph
3		nfcvd 2320	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2334	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2333	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1568	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1761	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1475	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   F/wnf 1460   E.wex 1492   e. wcel 2148   F/_wnfc 2306

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  nfneld  2450  nfraldw  2509  nfraldxy  2510  nfrexdxy  2511  nfreudxy  2650  nfsbc1d  2979  nfsbcd  2982  sbcrext  3040  nfsbcdw  3091  nfbrd  4046  nfriotadxy  5834  nfixpxy  6712