Theorem nfcrii 2367

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 2366	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	1, 2	ax-mp 5	. . 3 |- F/ x z e. A
4	3	nfri 1567	. 2 |- ( z e. A -> A. x z e. A )
5	4	hblem 2339	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1395   F/wnf 1508   e. wcel 2202   F/_wnfc 2361

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363

This theorem is referenced by:  nfcri  2368