Theorem nf4 1868

Description: Variable x is effectively not free in ph iff ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

Ref	Expression
nf4	|- ( F/xph <-> (A.xph \/ A.x -. ph))

Proof of Theorem nf4

Step	Hyp	Ref	Expression
1		nf2 1866	. 2 |- ( F/xph <-> (E.xph -> A.xph))
2		imor 401	. 2 |- ((E.xph -> A.xph) <-> (-. E.xph \/ A.xph))
3		orcom 376	. . 3 |- ((-. E.xph \/ A.xph) <-> (A.xph \/ -. E.xph))
4		alnex 1543	. . . 4 |- (A.x -. ph <-> -. E.xph)
5	4	orbi2i 505	. . 3 |- ((A.xph \/ A.x -. ph) <-> (A.xph \/ -. E.xph))
6	3, 5	bitr4i 243	. 2 |- ((-. E.xph \/ A.xph) <-> (A.xph \/ A.x -. ph))
7	1, 2, 6	3bitri 262	1 |- ( F/xph <-> (A.xph \/ A.x -. ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357  A.wal 1540  E.wex 1541   F/wnf 1544

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-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)