Theorem nfcrii 2341

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 2340	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	1, 2	ax-mp 5	. . 3 |- F/ x z e. A
4	3	nfri 1542	. 2 |- ( z e. A -> A. x z e. A )
5	4	hblem 2313	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1371   F/wnf 1483   e. wcel 2176   F/_wnfc 2335

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  nfcri  2342