Theorem nfcrii 2483

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

Hypothesis

Ref	Expression
nfcri.1	|- F/_xA

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/_xA
2		nfcr 2482	. . . 4 |- ( F/_xA -> F/x z e. A)
3	1, 2	ax-mp 5	. . 3 |- F/x z e. A
4	3	nfri 1762	. 2 |- (z e. A -> A.x z e. A)
5	4	hblem 2458	1 |- (y e. A -> A.x y e. A)

Syntax hints:   -> wi 4  A.wal 1540   F/wnf 1544   e. wcel 1710   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfcri  2484