Theorem nfcrii 2325

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcri.1	|- F/_ x A

Assertion

Ref	Expression
nfcrii	|- ( y e. A -> A. x y e. A )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem nfcrii

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcri.1	. . . 4 |- F/_ x A
2		nfcr 2324	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	1, 2	ax-mp 5	. . 3 |- F/ x z e. A
4	3	nfri 1530	. 2 |- ( z e. A -> A. x z e. A )
5	4	hblem 2297	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1362   F/wnf 1471   e. wcel 2160   F/_wnfc 2319

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 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321

This theorem is referenced by:  nfcri  2326