Theorem nfcrii 2274

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 2273	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	1, 2	ax-mp 5	. . 3 |- F/ x z e. A
4	3	nfri 1499	. 2 |- ( z e. A -> A. x z e. A )
5	4	hblem 2247	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1329   F/wnf 1436   e. wcel 1480   F/_wnfc 2268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270

This theorem is referenced by:  nfcri  2275