Theorem r19.3rz 3641

Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Hypothesis

Ref	Expression
r19.3rz.1	|- F/xph

Assertion

Ref	Expression
r19.3rz	|- (A =/= (/) -> (ph <-> A.x e. A ph))

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem r19.3rz

Step	Hyp	Ref	Expression
1		n0 3559	. . 3 |- (A =/= (/) <-> E.x x e. A)
2		biimt 325	. . 3 |- (E.x x e. A -> (ph <-> (E.x x e. A -> ph)))
3	1, 2	sylbi 187	. 2 |- (A =/= (/) -> (ph <-> (E.x x e. A -> ph)))
4		df-ral 2619	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
5		r19.3rz.1	. . . 4 |- F/xph
6	5	19.23 1801	. . 3 |- (A.x(x e. A -> ph) <-> (E.x x e. A -> ph))
7	4, 6	bitri 240	. 2 |- (A.x e. A ph <-> (E.x x e. A -> ph))
8	3, 7	syl6bbr 254	1 |- (A =/= (/) -> (ph <-> A.x e. A ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544   e. wcel 1710   =/= wne 2516  A.wral 2614  (/)c0 3550

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-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  r19.28z  3642  r19.27z  3648